No this isn't the reason why your tail lights don't work. Tail-lights and the turn signal are two different systems in the car: if your turn signal is broken that doesn't necessarily mean that your tail-light will be broken.
fuse relay wiring switch
Replace the turn signal flasher relay.
The brake lights and signal lights are tied together in the signal switch, also known as the "multifunction switch".
it dont have lights
bad fuse or hazard switch
The fog light switch is on the end of the turn signal switch. Just pull the end of the turn signal switch and the fog lights will come on. (Turn signal switch and fog light switch are one assembly)
Since most brake/turn use the same filament in the bulb(1157) then if the turn signal switch is bad or offset then it will effect the brake lights.
the multifunction switch... ( turn signal) twist it to turn on the lights and once the lights are on pull it towards the door to turn the fog lights on
Pulling the turn signal switch toward you should switch the lights to the low beams. If that doesn't work, you may have to replace the switch, which means replacing the entire turn signal stalk.
If you have the same bulb for brake and signal, brake light power needs to be interrupted by the signal lights, The brake circuit has to go through the signal light switch so the problem could be there.
Does your hazard lights only work when you turn the flashers on and the turn signal on at the same time? GM is known for having a multi-purpose switch... this switch controls your lights, turn signals, hazard lights and high beams... sometimes cruise control too... well anywho, sometimes that switch is partially broken... I recommend hooking a multi-meter to the wires that hook the multi-switch up and test all the functions of the switch... but I guarantee if you put a new multi-switch on the car that your hazards lights will function properly... I also recommend buying the part through an after-market company because it is cheaper...
It is the turn signal switch combined with the head lights, wipers, etc.