High beams on when left turn signal is activated, 08 GMC Envoy

Does it do the same with the lights on 'High' beam? If it works correctly on 'High' beam it could be the switch in the lever at fault.
Next check the fuses for the turn signals and the Hazard lights. They will be in the fuse box under the hood. Some vehicles will have separate fuses for Left, Right and Hazard lights others may share fuses.
If the fuses are OK it could be a faulty flasher unit. Check to see if the hazard lights work correctly. Some vehicles use separate flasher units for turns and hazards and others share 1.

Did you test the low beam electrical circuit using a volt meter ? Look at a wiring diagram to see what all is involved in turning the lights on ? Do you know how to test electrical circuits ?
Also reading service repair manual about the head lamps opp. would be a good thing to do .
Headlamps
The major common components of this system are the body control module (BCM), ambient light sensor, multifunction switch, park brake switch, and the headlamps.
The headlamps may be turned on in 2 different ways. First, when the driver places the headlamp switch in the HEADLAMP position, normal operation occurs. Second, with the headlamp switch is placed in the AUTO position, automatic lamp control (ALC) occurs. During ALC control, the headlamps will be in daytime running lamp (DRL) operation in daylight conditions, or low beam operation in low light conditions.
The low beam relay and high beam relay receive battery positive voltage directly from the battery through the HDM and COILS fuses in the underhood fuse block. The BCM supplies a ground signal to the low beam relay coil for automatic headlamp control and normal headlamp operation. The BCM also supplies ground to the high beam relay coil for high beam headlamp operation. When the driver places the headlamp switch in the HEADLAMP position and the dimmer switch is in the low beam position, the low beam relay energizes sending current flow through the left and right low beam headlamp fuses to both low beam headlamps. The dimmer switch sends a ground signal to the BCM in the high beam position and a momentary ground signal in the flash-to-Pass (FTP) position from G102 to deactivate the low beam headlights and activate the high beam headlights. With the headlights in the low beam position, the high beams may be momentarily turned on or flashed by activating the FTP portion of the switch.

Low Beams - Standard
The standard low beam headlamps receive voltage from the headlamp driver module located in the underhood fuse block. The body control module (BCM) controls the headlamp driver module with the headlamp low beam relay control circuit. Low beam control is determined by a signal on the headlamp dimmer switch signal circuit from the multifunction switch. When the headlamps are requested ON and the headlamp dimmer switch signal circuit is open, the BCM sends a pulse width modulated (PWM) ground signal though the headlamp low beam relay control circuit to the headlamp driver module which illuminates the low beam headlamps at a 100 percent intensity level. The right and left low beam circuits are independent and are fused separately in the underhood fuse block. The LOW HDLP-RT 10A fuse supplies voltage to the right low beam headlamp and the LOW HDLP-LT 10A fuse supplies voltage to the left low beam headlamp. The right and left headlamps are grounded at G103 and G107 respectively.

Your best bet mite be to have a qualified technician check it

Use a test light at the connector, to the low beam lamp. Test for power and ground. If both wires are working then the issue is either the bulb or the connection. If testing the wires fails then it will need to be traced back to the source. When you activate the headlights the BCM receives the signal and sends power to either the low or high beam relay and from the relay to the appropriate lamp. If your looking for some other specific piece of information please write back. Good Luck.

Your best bet , take your vehicle to a qualified repair shop . Before you want to replace the BCM . . Professional tech would check for DTC'S - diagnostic trouble code's in the BCM before replacing part's that are not needed .

No signal at signal wire on relay ???? Which circuit wire mite that be ? Hooking up a factory or professional scan tool to view input data from the switch would also be a good step .

Headlamps
The headlamps may be turned ON 2 different ways. First, when the driver places the headlamp switch in the HEADLAMP position, for normal operation. Second, with the headlamp switch placed in the AUTO position, for automatic lamp control (ALC). During ALC, the headlamps will be OFF in daylight conditions, or low beam operation in low light conditions. Battery positive voltage is supplied to the low beam relay and the high beam relay in the underhood fuse block. When low beam headlamps are requested, the body control module (BCM) supplies ground to the coil side of the low beam relay. This activates the relay and the relay supplies voltage to the left hand low beam headlamp through the LH LOW BEAM fuse and to the right hand low beam headlamp through the RH LOW BEAM fuse. Ground for the left hand low beam headlamp is provided at G101. Ground for the right hand low beam headlamp is provided at G102. When high beam headlamps are requested, the body control module (BCM) supplies ground to the coil side of the high beam relay. This activates the relay and the relay supplies voltage to the left hand high beam headlamp through the LH HIGH BEAM fuse and to the right hand high beam headlamp through the RH HIGH BEAM fuse. Ground for the left hand high beam headlamp is provided at G101. Ground for the right hand high beam headlamp is provided at G102.

Headlamps Inoperative - Low Beams
Step
Action
Yes
No
Schematic Reference: Headlights/Daytime Running Lights (DRL) Schematics
Connector End View Reference: Master Electrical Component List
1
Did you perform the Diagnostic System Check - Vehicle?
YES - Go to Step 2
NO - Go to Diagnostic System Check - Vehicle
2
Place the headlamp switch in the HEADLAMP position.
Place the headlamp dimmer switch in the LOW position.
Do the low beam headlamps illuminate?
YES - Go to Testing for Intermittent Conditions and Poor Connections
NO - Go to Step 3
3
Disconnect the turn/headlamp switch connector C2.
Connect a 15-amp fused jumper between the headlamp low beam signal circuit and a good ground.
Do the low beam headlamps illuminate?
YES - Go to Step 4
NO - Go to Step 5
4
Inspect for poor connections at the harness connector of the turn/headlamp switch. Refer to Testing for Intermittent Conditions and Poor Connections and Connector Repairs .
Did you find and correct the condition?
YES - Go to Step 7
NO - Go to Step 6
5
Repair an open or high resistance in the headlamp low beam signal circuit. Refer to Circuit Testing and Wiring Repairs .
YES - Did you complete the repair?
NO - Go to Step 7
--
6
Replace the turn/headlamp switch. Refer to Turn Signal Multifunction Switch Replacement .
YES - Did you complete the replacement?
NO - Go to Step 7
--
7
Operate the system in order to verify the repair.
Did you correct the condition?
YES - System OK
No - Go to Step 2

It's more then likely the dimmer switch .

DTC B2575
Circuit Description
The body control module (BCM) turns on the LOW BEAM relay when the ambient light sensor senses a dark condition or the low beam headlamps are requested ON. Battery positive voltage is supplied to the LOW BEAM relay at all times. The BCM grounds the LOW BEAM relay control circuit in order to activate the relay.
DTC Descriptor
This diagnostic procedure supports the following DTC:
DTC B2575 Headlamp Control Circuit

B2575.01
Will set if there is a short to voltage in the low beam relay control circuit.
B2575.06
Will set if there is an open or short to ground in the low beam relay control circuit.

Check that the bulbs actually work, with an external power supply.

bad earth on the bulb holder or the wiring, usually a good clean should sort it out.

I would sugest a bad earth, 1st remove the brake globes & see if the sockets that house them are rusty, if so clean with emery paper & refit globes. See if this works 1st. ps don't forget to clean the globes contacts to

Most likely your problem is under the steering wheel. What is where the switch for the turn signals is. No high beams makes me think someone may have had the steering column out and the high beam switch is out of adjustment or the rod that controls that is out of its socket.
I do not know for sure but with the car being so new it may be controlled by the computer too. I left GM in 2001 and not up 100% on todays advancement in the electrical system.

my oldsmobile 1995 cutlass supreme has the same problem, so I replaced turn signal switch.

For the I6 engine there is a relay for low beams, it's relay # 46. Switch it with relay #45 (runs the fan, same model relay) and see if your lights now come on. These 2 relays are the only 2 black ones in there (part # 15016745). Also, try to clean the contacts very well on the fuse box and the relay. If not, you may have some frayed wiring in the harness.

By the way, the high beam relay is #43, the gray one next to the windshield wiper relay #42.