Turn signals light but do not flash

These vehicles were known for faulty hazard switches. They would cause an intermittent turn signal flash when you hadn't activated turn signals or they would prevent the turn signals from working at all. Operate the hazard switch and see if the turn signals work again. If so, replace the hazard switch

You might possibly have a bad flasher module.

Turn Signal Lamps
The LT/TRN/SIG and RT/TRN/SIG fuses in the I/P fuse block supplies battery positive voltage to the body control module (BCM). When the turn signal switch is placed in either the left or right position, a signal is sent to the BCM to illuminate the desired turn signal lamps. The BCM has an internal flasher that 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, a separate ground signal is sent to the BCM to activate and flash all turn signal lamps including both IPC turn indicators. The front and rear turn signals are grounded at G401.
The IPC illuminates the TURN SIGNAL ON indicator in the driver information center (DIC) when the IPC determines that the turn signal is active for more than 1.2 km (0.75 mi). The IPC cluster sends a serial data message to the radio in order to activate the audible warning.

Your hearing it coming from the diver side front speaker . There is no relay , the BCM - body control module makes the turn signals work . What problem are you having ?

Turn Signal Lamps
The FRT TRN/HAZ/CRTSY/MIRR/CARGO and REAR TURN/HAZ fuses in the underhood fuse block supplies battery positive voltage to the body control module (BCM). When the turn signal switch is placed in either the left or right position, a signal is sent to the BCM to illuminate the desired turn signal lamps. The BCM has an internal flasher that 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, a separate ground signal is sent to the BCM to activate and flash all turn signal lamps including both IPC turn indicators. The front and rear turn signals are grounded at G105.
The IPC illuminates the TURN SIGNAL ON indicator in the driver information center (DIC) when the IPC determines that the turn signal is active for more than 1.2 km (0.75 mi). The BCM sends a class 2 message to the radio in order to activate the audible warning.

Your best bet , take it to a qualified repair shop.

Generally speaking, symptoms of a bad or failing turn signal bulb are usually the culprit...in your case, either it's the front left or rear bulb. Varies for different vehicle makes...common signs include:

1. turn signal light staying on solid
2. or blinking very fast
3. or either sides of the bulbs stop blinking altogether.

To test this, make sure you reconnect everything you disconnected.
Turn the left signal light on; go to the front of the vehicle, is the turn signal bulb illuminated? If it lights up, even though it is not blinking obviously the bulb is okay. If it doesn't, replace it- problem solved. You'll do the same procedure for the rear turn signal Is that lit up? If not replace the bulb and the problem is solved.
Another common failure is when the turn signal blinks really fast, same procedure as mentioned above.
If both bulbs for the left side and right side illuminate but don't blink, the problem is in the steering column turn signal switch.
If the left or right side blinkers don't work properly at all...then check the integrated hazard warning/turn signal flasher relay- that may be defective; again, that's usually the case if both sides of the signal light don't work.
If you are blowing any fuses related to the flasher system, you have an electrical overload in the turn signal circuitry.

Turn Signal Lamps
The turn signal lamps may only be activated with the ignition switch in RUN or BULB TEST position. The T/SIG fuse supplies ignition voltage to the turn signal switch. When the turn signal switch is placed in either the left or right position, the turn signal switch sends an ignition voltage to the hazard lamp/turn signal flasher, which sends an on-off signal to the turn signal switch. When the turn signal switch receives the on-off signal it will send that signal to the following lamps on the LH or RH side:
?€¢
The Rear Turn Lamp

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The Turn Signal Indicator

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The Sidemarker Lamp

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The Sideview Mirror (Pontiac Thunder only)

The turn signal switch also sends a separate on-off signal to the Daytime Running Lamps (DRL) control module, which relays the signal to the appropriate front turn signal. The front turn signals are grounded at G101 and G102. The rear turn signals are grounded at G401. The sidemarker lamps are grounded through the corresponding front parklamp.
The sideview mirror turn signals are a bank of LEDs placed behind the mirror glass.
The hazard lamps operate on battery voltage. The HAZARD fuse supplies battery voltage to the turn signal switch. When the hazard switch is pressed, the turn signal switch sends battery voltage to the hazard lamp/turn signal flasher, which sends an on-off signal to the turn signal switch. The turn signal switch send the on-off signal to all the turn signal lamps, causing all the lamps to flash including both IPC turn signal indicators. The hazard lamp/turn signal flasher is grounded at G200.
When the parklamps are ON, the sidemarker lamps receive voltage from the park lamp supply voltage circuit and illuminate. The sidemarker lamp is grounded through the corresponding rear turn signal. When the turn signal switch is place in the left or right hand position, the rear turn signal illuminates and the sidemarker lamp will loose its ground, causing the sidemarker lamp to turn off when the turn signals are on and to illuminate again when the turn signal is off.
Which blub an socket did you replace ? check the other blub ? The switch and or the flasher module can cause this problem .
Replace the turn signal switch. Refer to Multifunction, Turn Signal, and Hazard Switch Replacement - On Vehicle in Steering Wheel and Column.
Did you complete the replacement?

Which

change flasher relay in fuse box

That is the hazard lights! Many automobiles use their front and rear signal lights to create hazard lights in a distinctive lighting pattern which can be used to alert other drivers to a problem. Typically, hazard lights run on the same circuit as regular lights, although they are controlled with a separate switch. Knowing how and when to use hazard lights can be useful in an emergency situation, although you should also be equipped with highway flares. When the switch is activated, all of the turn signals on the vehicle will simultaneously illuminate and start flashing in a rhythmic pattern. This pattern is highly visible and very unique, so that drivers will not confuse it with turn signals or approaching headlights.

The rear left turn signal bulb must be replaced immediatly before it melts the light socket, if it has'nt yet. Also please verify that the correct light bulb is being installed. because even though many bulbs look the same they are not identical. Installing the wrong bulb can be the begining of many, many problems in the particular circuit

Ok. First of all, chk all the bulbs. Well at least the four parking lamps. The ones that are supposed to flash. One or two are burnd out. It happens. The flasher(s) work on resistance and that is provided by the bulbs.Brake likes and the turn signals are electrically associated.and you already said one side does not work, so let's start with bulb inspection Let me know