Turn signal clicking.. not the hazard switch!
 

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I just HAVE to ask. You have checked your passenger side bulb right? Also, what year monte?

passenger side bulb works with hazards on front and brake side. Even when the turn signal lever is to the right the interior green turn signal does not turn....

Sorry if I missed it, but what year is your Monte?

This has all the signs of a bulb being burnt out. When a turn signal bulb is burnt, the flasher typically flashes quickly to let you know there's a bulb out (before the cops do). It can be any one of the bulbs used by the right turn signal. You will need to do a visual check on all of them.

Sometimes comparing which bulbs light up with the left turn signal with the right side helps. Don't forget to check the side marker bulbs too. You also shouldn't assume that a bulb is good because the parking lights light up. Some bulbs have two filaments in them. The parking lights use a different filament than the turn signal and brakes.

That's the basics. But there's another possibility.

Do you have aftermarket LED bulbs installed? LED's put a different load on the electrical system. And some LED's cause a problem. If the car came from the factory with incandescent bulbs, it's looking for the same resistance as an incandescent bulb. If you install LED's, you're putting a different resistance load on the system. And the car thinks you have a bulb burnt out.

I have a 2000 monte carlo.

The right turn signal works in the front and the back when the hazards are on. However... when the right turn signal is turned on neither the front nor rear turn signals light up.

Hazard button pushed:
Right green flasher on dashboard: WORKS. Right front exterior turn signal: WORKS. Right Rear exterior turn signal: WORKS. NORMAL CLICKING NOISE.

Right turn signal lever put into right turn signal on position:
Right green flasher on dashboard: OFF. Right front exterior turn signal: OFF. Right rear exterior turn signal: OFF. FAST CLICKING NOISE.

Could the bulb still be burned out in the front if it works with the hazards?

Also... I do not have any aftermarket lights.

I would go ahead and Replace the taillight, worst case scenario you are out 3 dollars.

Some vehicles I have encountered have had a fuse for the left and right turn signals separate, I don't know if 2000s do or not just a thought. It could also be a defective relay under the steering column I suppose (one of the little black boxes in the circuit board).

If none of that helps I hope you have a light tester because you will probably have to track that hot wire the length of the car.

Fixed guys. It was the turn signal switch. Expensive... but is done now. Thanks!