Pontiac firebird 2001, my turn signals don't work.

Bulbs burned out, Flasher bad or the turn signal switch under the steering wheel.

Yes, it does have a flasher, although it may be a combined unit with the hazard flashers relay. So you hear the clicking of the flasher relay? That seems to indicate the relay is working properly. Have you checked the bulbs and the turn signal fuse? Any turn signal bulbs light up anywhere, front or back?
How about your brake lights, do they work? Your car may have separate, stand alone turn signals in the rear, but many American cars use the same bulb and same filament on the bulb for the brake lights and turn signals. If your merc is like this and the brake lights work, then the bulb must be good.
Turn on the hazard flashers and see if all bulbs work and flash at each corner of the car. This would be the same bulb as for the turn signals. If they work, it could be a bad turn signal switch, may even be the flasher relay somehow malfunctioning, and in a few cases, a bad hazard switch can affect the turn signal switch. Check fuses and bulbs first. If you have any type of body control module that controls lights and other accessories, you may have to investigate that. I know I'm not being very helpful, just throwing out a few suggestions. Good luck.

There are two seperate flashers, one for the hazards, and one for the turn signals. They are located behind the lower instrument panel below the steering wheel. Turn flasher should be the one on the right. These are also notorious for the front light sockets to corode or burn out. check their condition and be sure to use connector lube when replacing bulbs there.

did u change the correct flasher?there are 2 flashers,the regular turn signal one and the 4way flashers,make sure u changed the right one,,,

if 4way flashers work & no signal,,,i would replace signal flasher,,,there are two flashers ,,,,

Check all the grounds for the running lights

Sounds like the flasher unit to me. Normally the emergency flasher and turn signal are separate.

The turns,hazards, high/ low headlamps, possibly cruise cont. and wipers, are all operated by the "multifunction switch". Replace it and you should be good to go. I hope this helps.

You have two separate flasher devices on your car. One operates the flashers, and the other, usually located near the fuse area, operates the turn signals. A trip to your local Autozone or Advance Auto Parts, or for that matter most reputable parts house will give the exact location of turnsignal flasher, and they can supply a new one.

Take the panel under the steering wheel off and its behind a little lip on the right side (left side of the stereo)

Its real easy to get too after you take that panel off.