My brake, back turn signal lights stop working and the top brake light stays on.

Have you checked the bulbs already?
Don't know what kind of car you are dealing with here.
Most vehicles - the brake lights and rear turn signals share the same electrical circuit to the rear tailights. Usually the turn signal switch will separate which signal goes to the back.
Do your brake lights work? Is there a High mounted stop lamp and does it work?
I have seen a Hazard Switch be ON and fuse blown causing the Hazard Flasher to be inop - which prevented the turn signals from working. Turning OFF the hazard flasher brought back the T/S.
Check the rest of the lights - see what does and doesn't work - and do the front turn signals flash? Stay On? Or flash really slow?
Let me know what you find out and if this helps you at all.

I have to believe that with all of the "non related" issues happening that there is a wiring harness problem. If you track the harness back from the tail lights you will probably find a nick or a break in the line. If nothing shows then check the connection point which should be somewhere around your fuel tank. It will be a male/female fitting that will pull apart and you can check at that point with a meter if it is working to that point. If not it then you will have to work your way all the way up to find the fault point.

I think you have a bad ground at the back on both rear lights or their earth out on the same connection but these american cars use a funny system where the stop lights are the indicators and not a seperate yellow light like europeon motors ,Now these brake light switches also allow the indicators to work by shutting off the feed to the stop light so it flashes with indicators ,Why american cars use this system i dont know as its so complicated for the stop light switch function but try the ground or earth wire on the back lights and if no joy then ask a american mechanic with experiance of this sort of circuitry as when i get american cars for import i have to fit new back lights and disconnect the brake light switch .

IF THE FUSES INSIDE THE CAR & UNDER THE HOOD HAVE ALL BEEN CHECKED WITH A GOOD FUSE TESTER, & THE BRAKE LIGHT SWITCH REPLACED, THE TURN SIGNAL SWITCH IS NEXT. THE BRAKE LIGHTS GO THROUGH THE TURN SIGNAL SWITCH SO THEY CAN BE USED AS TURN OR BRAKE LIGHTS. I'M ASSUMING THE BULBS ARE OK. REMOVE THE PLASTIC FROM AROUND THE TURN SIG. SWITCH. STAY AWAY FROM ANY WIRES THAT ARE IN THE YELLOW SLEEVE. THEY ARE AIR BAG WIRES. WITH BRAKE PEDAL DEPRESSED, THERE SHOULD BE POWER IN THE WHITE WIRE. IF THERE IS NO POWER IN THE WHITE WIRE, YOU MAY HAVE TO PUT A BYPASS WIRE FROM THE WHITE WIRE COMING OUT OF THE BRAKE LIGHT SWITCH TO THE WHITE WIRE GOING INTO THE TURN SIGNAL SWITCH. IF THAT DOESN'T MAKE THE BRAKE LIGHTS WORK, THE TURN SIGNAL SWITCH MUST BE BAD.

Brake light on top works tells me the switch at the pedal is ok. Turn signals working tells me that the turn signal filaments have power.

I would try changing the bulbs and if that doesn't work have an electrical shop trace the wires and check for power on the brake light circuit in the back.

The brake light switch is located is located under the
dash, at top of the brake peddle.
To test the sw, use a 12v test light check for constant power out of the switch, with nothing else on. If there's pwr. out replace the switch.
Hope this helps. Good luck

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

THE PROBLEM IS IN THE MULTIFUNCTION SWITCH, REPLACE IT.

your problem is attributed to the turn signal switch , both the left and right turn signals go thru the switch as well as the brake lights , it is a common problem with these cars.