Left turn signal stops flashing when brake pedal is depressed.

What's a DHS ?
Stop Lamps

The STOP LP fuse in the rear fuse block supplies battery positive voltage to the normally open stop lamp switch. When the brake pedal is pressed, the stop lamp switch contacts close and battery positive voltage is supplied to the center high mounted stop lamp (CHMSL) and the turn signal/hazard flasher module stop lamp switch signal input. In response to this input the turn signal/hazard flasher module supplies battery positive voltage to the left and right LED tail lamp assemblies. The CHMSL and left tail lamp have ground at G402. The right tail lamp has ground at G401. The stop lamps illuminate when they receive both power and ground.
es battery positive voltage to the left and right LED tail lamp assemblies. The CHMSL and left tail lamp have ground at G402. The right tail lamp has ground at G401. The stop lamps illuminate when they receive both power and ground.

Turn signal switch is probably the problem ,but without testing that is just a guess. Do know i replaced a few on grand am's . What relay ???? There is no relay in the fuse box for the turn signals . Your vehicle has turn / hazard flasher module .
Turn Signal Lamps
The TURN-B/U fuse and the STOP/HAZ fuse in the IP fuse block, supplies battery positive voltage to the flasher module. The turn signal lamps may only be activated with the ignition switch in the ON position. When the turn signal switch is placed in either the left or right position, current flow is from the flasher module's output to the turn signal switch assembly. The flasher module then sends an on-off voltage signal to either the left or right turn signals and their instrument panel cluster (IPC) indicator. When the hazard switch is pressed, all the turn signal lamps will flash including both IPC turn signal indicators.
Located under the driver side of the dash above the brake pedal ,on a metal bracket .
Turn Signal/Hazard Flasher Module
On the dash harness near the steering column

Stop Lamps
The brake pedal position sensor is a resistive device used to sense the action of the driver's application of the brake pedal. The brake pedal position sensor provides an analog signal that will increase as the brake pedal is pressed.
The brake pedal position sensor receives both a 5-volt reference voltage and a low reference signal from the electronic brake control module (EBCM). When the brake pedal is pressed, a variable signal is applied from the brake pedal position sensor to the EBCM through the brake sensor signal circuit. The fuse block left rear supplies battery positive voltage to the bass relay switch through the BASS fuse. One side of the bass relay coil is permanently grounded at G402. The EBCM supplies voltage to the bass relay coil upon receiving the signal from the brake pedal position sensor. The switch contacts of the bass relay close and battery positive voltage is applied through the bass relay switch to the center high mounted stop lamp (CHMSL), and to the turn signal/hazard flasher module through the stop lamp switch signal circuit. The ground for the CHMSL is applied from G402. The turn signal/hazard flasher module then supplies voltage to the left and right stop lamps in the tail lamp assemblies that have ground at G401 and G402. For export vehicles the stop lamps receive battery positive voltage directly from the bass relay.

Do you know where the rear fuse box is ? Check BAS fuse 15amp. and the bas relay , should be marked on the inside of the fuse box lid . The rear fuse box is located

Fuse Block - Left Rear
In the passenger compartment, under the left side of the rear seat at carpet
Power and Grounding Component Views
Electrical Center Identification Views
Fuse Block - Right Rear
In the passenger compartment, under the right side of the rear seat at carpet.

If fuse and relay are ok your best would be to take to a cadillac dealer .

Kevin , this is a very complex way of stoplamp control , I will post GM service Description and Operation of how they work an let you deside if you want to tackle this ! Do the center high mount brake lights work ?
The brake pedal position sensor is a resistive device used to sense the action of the driver's application of the brake pedal. The brake pedal position sensor provides an analog signal that will increase as the brake pedal is pressed.
The brake pedal position sensor receives both a 5-volt reference voltage and a low reference signal from the electronic brake control module (EBCM). When the brake pedal is pressed, a variable signal is applied from the brake pedal position sensor to the EBCM through the brake sensor signal circuit. The fuse block left rear supplies battery positive voltage to the bass relay switch through the BASS fuse. One side of the bass relay coil is permanently grounded at G402. The EBCM supplies voltage to the bass relay coil upon receiving the signal from the brake pedal position sensor. The switch contacts of the bas relay close and battery positive voltage is applied through the bas relay switch to the center high mounted stop lamp (CHMSL), and to the turn signal/hazard flasher module through the stop lamp switch signal circuit. The ground for the CHMSL is applied from G402. The turn signal/hazard flasher module then supplies voltage to the left and right stop lamps in the tail lamp assemblies that have ground at G401 and G402. For export vehicles the stop lamps receive battery positive voltage directly from the bass relay.

There could be a code stored in the EBCM - electronic brake control module - DTC B3903 Stop Lamp Relay Circuit
DTC C0278 Brake Pedal Position Sensor Not Calibrated
there are several other codes that pertane to the brake lights . you could try Replaceing the bas relay !
Fuse Block - Left Rear
In the passenger compartment, under the left side of the rear seat at carpet.

YOU MIGHT HAVE A BURNT OUT RELAY IN FUSE COMPART UNDER YOU CARS HOOD CHECK THAT OUT THRU YOUR CARS MANUAL,IT WILL SHOW YOU WHICH RELAY IT IS?

It will turn on when I hit the button and the turn signals do work. All of this has led to my ABS lights to come on as well. ??????????? Your best bet , take it to a qualified repair shop or learn to do better explanation's ! Or lean to do automotive electrical testing .
Basic Electricity for Service Techs Ohm law Current Flow Opens Shorts
Electric Testing Techniques You Need to Know

How to Use Multimeter to Troubleshoot Common Problems

Do you know what a wiring diagram is ? How to use a DVOM - digital volt ohm meter ? Free wiring diagrams here http://www.bbbind.com/free_tsb.html Enter vehicle info. year make ,model and engine size. Under system click on lighting ,then under subsystem click on turn signals . Click the search button then the blue links .Check power an ground circuits ! This is how we diagnose electrical faults , not just replacing parts .

Turn Lamps
Voltage is applied through the TURN fuse and the turn/hazard flasher when the ignition switch is in the following positions:
• RUN
• BULB TEST
• START
The voltage travels to the contact in the hazard switch. The hazard switch is located in the turn/hazard-headlamp switch assembly. This contact is normally closed.
When the left turn signal is turned on, voltage is applied to the following components.
• The left turn indicator
• The left front park/turn lamp circuit 14
• The left rear turn lamp circuit 18
The turn lamps turn on immediately. The current flow heats up the timing element. The timing element is located in the turn/hazard flasher. The timing element repeatedly opens and closes the circuit. This action causes the turn lamps to begin flashing.
The voltage that is applied to the left front park/turn lamp is also applied to the left front marker lamp. If the headlamp switch is in the OFF position, the left front marker lamp will find a path to ground through one of the following lamps. The following lamps provide low resistance paths to ground:
• The right front marker lamp
• The right turn indicator
• The right front park/turn lamps
• The left front park/turn lamps
The left front marker lamp flashes with the turn lamps. The voltage drop across the marker lamp is much higher than the voltage drop across the other lamps. The lamps that are used for the ground path will not flash.
When the headlamp switch is in either the PARK or HEAD positions, voltage is applied through the following components:
• The EXT LP fuse
• The marker lamps
• The park lamps
When the left turn signal is turned on, the left marker lamp will have voltage at both connections. The left marker lamp does not light. When the turn/hazard flasher stops the voltage to the turn lamps, the marker lamp is grounded through the turn lamp. The marker lamp does not go on. The left front marker lamp flashes in the following manner:
• The left front marker lamp flashes on when the turn filament of the left front park/turn lamp goes off.
• The left front marker lamp flashes off when the turn filament of the left front park/turn lamp goes on.
Voltage is applied to the right lamps in the same manner, when the right turn signal is turned on.
Hazard Lamps
Voltage is applied at all times through the following components to the normally open contact of the hazard switch in the turn/hazard-headlamp switch assembly:
• The STOP HAZ fuse
• The turn/hazard flasher
When the hazard switch is turned on, voltage is applied to the following components:
• The front turn lamps
• The rear turn lamps
The following lamps flash on and off:
• All of the turn lamps
• Both of the turn indicators
The front marker lamps flash with the hazard lamps on, just as the front marker lamps flash when the turn lamps are on. If the headlamp switch is in the OFF position, the front marker lamps flash on when the hazard lamps are on. The front marker lights will flash on when the hazard lamps are on if the headlamp switch is in one of the following positions:
• PARK
• HEAD
When the hazard lamps are on, the following conditions apply:
• The turn circuit is always open.
• The turn/hazard flasher controls the lamp.
Park/Marker Lamps
Voltage is applied through the EXT LP fuse, to the headlamp switch at all times. Voltage is applied to the park lamps and the marker lamps when the headlamp switch is in the following positions:
• PARK
• HEAD
The front marker lamps are grounded through the turn filament of the respective front park/turn lamp. The front marker lamps light as a result.
Stop Lamps
Voltage is applied through the STOP HAZ fuse to the brake switch at all times. When the brake pedal is depressed, the brake switch closes. This action applies voltage through the turn/hazard-headlamp switch assembly to the following lamps:
• The high mounted stop lamp assembly
• The left tail/stop-turn lamps
• The right tail/stop-turn lamps

The flasher does not stop the brake lights from working. Check # 18 fuse in fusebox under lower left side of dash. if it's ok, the brake light switch, on the brake pedal arm is next. light green wire with red stripe is hot if fuse 18 is ok. Depress brake pedal, light green with no stripe should have power. if not, replace brake light switch. If light green wire has power with pedal depressed, combination switch under the steering is next. Remove the cover from around the combo switch {under the steering wheel}. depress brake pedal, light green wire should have power. If it does, pedal still depressed, orange with light blue wire & wire with light green & orange should have power. Those 2 wires are brake & rear turn signal wires. { The turn signal switch should be in the middle, not right position or left. while checking for power in the last 2 wires. Good Luck drvolt.

change flasher relay in fuse box

Probably the flasher, as I think you have a separate flasher for the turn signals under the dash above the brake pedal. It is probably next to the hazard flasher. If you can't find it, turn the hazards on and follow the sound. The turn flasher is probably next to the hazard flasher. Please let me know if this isn't the problem or you have more questions.

check the turn signal switch