Why is the left turn signal flashing quickly? Left brake light not working

Has the vehicle been in an accident on the left rear side. First check the bulb to be sure it is not defective, trade it out with the side that works properly to test. If the bulb works well then My thought are you may have a shorted wire. The turn signal flasher works by heat build up within the flasher the heat is caused by the load of the bulbs, the load heats the metal spring causing it to open the circuit then as quickly as it cools down the circuit closes again lighting the bulbs.
There is also a possibility of a defect in the turn signal mechanism on the steering column. The turn signal switch on many models does three different things on many models that use the same bulb element / or bulb for brake and turn on rear of vehicle. It controls the circuit for brake light by sending power both left and right side rear when turn signal is in neutral position then when turn signal is in left turn the switch cancels brake light power to left side rear and supplies flasher power to left side while maintaining brake light supply to the right side rear, and so on opposite when the right turn is chosen on turn signal switch. The turn signal indicators on the dash normally run off the same circuit as front turn signal bulb only which is also controlled by turn signal switch. Front has turn signals but no brake light.
I think there is an additional load on the left side rear turn circuit and it could be either a wire that is shorted or the turn signal switch if the same bulb element is used for both functions.
Often times if trailer lighting is connected the flasher will also blink a little faster because of the extra load of additional bulbs.

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

Fast flashing dash signals mean there is a bulb issue-DEFINATELY!

IF LEFT FRONT BLINKER NOT WORKING FRONT TURN SIGNAL BULB BURNED OUT.

Hello. The symptoms you are describing are likely due to a poor ground. Temporarily install a new ground wire by securing a piece of copper wire to the frame with a self tapping screw. Use this ground wire along with the hot wire to see if each of these lights works. If it does, trace the ground wire back to the frame, unscrew it and clean up the metal with emery cloth so you reestablish a good ground.

You may have some wiring 'grounding' problems in the areas of concern. Remove each connection, and clean any rust. Place a light film of grease and a fresh star washer next to ground point, then the wire connector and other washers. Tighten firmly. Check the lamp sockets for any corrosion and replace same if found. The front lamp flashes rapidly because the rear light not working causes the flasher unit to switch faster.

you need to replace the flasher unit

its your blinker box.. all major lights run through there

Usually when your signal blinks rapidly it indicates a blown bulb.   Switch on your four-way flashers and check your bulbs at each corner.  On the brake light staying lit, check to see if the left-rear bulb is installed correctly and is the correct bulb for the application.  I've seen people force bulbs in sockets causing the braking element of the bulb to light instead of the parking element.  The braking element is much brighter than the parking element.

Looks as though you have earthing problems.
I would make sure that the earthing is O.K on flasher and brake.
You would then find your symptom disappear.

Hope that sorts it.