Brake lights not working. Turn signal flashes quickly like it is burned out but still working. Fuses all look ok

Dead short

replace the right rear bulb. the quick flashing indicates a bad bulb. it could be a wiring problem to the rear for the signal light that you will have to check out starting at the socket and working forward.

Do turn signals work ok? Do you have current at the brake switch?

usually a ground problem in the harness or the lamp socket.

Lets start at the beginning.
The turn signals and brake lights use different fuses but both go thru the turn signal switch and use the same bulbs.
Have you checked the fuses for each circuit and does each switch have power in and out ?

Replace the turn signal switch. All of these lights run threw it.

faulty turn signal switch would effect brake lights if its faulty.because brake lights and turn signal feeding off same circuit.

your socket may not be grounded properly, I have seen this before, and the simplest way to fix is change the socket with a new one.Not another used one it may also be bad.