1993 Toyota Supra Loud thudding and jolting from the rear o

Hello LadySupra

Loud thudding bang noise from the rear and jolting forward, could be sign of engine misfiring.
I believe it is time for a tune up.
Specially pay more attention to how the spark plugs look like.
If the spark plugs are light brown or tan colored, they are good.
Black fluffy carbon deposits indicate an overly rich fuel mixture or possibly a weak spark can be cause by high resistance in plugs wires.
A wet spark plug means the plug has not been firing.
There could also be a vacuum leak.
Visually inspect all the vacuum hoses and connections. Look for disconnected, loose or cracked hoses, broken fittings.

Hello LadySupra

You probably have broken side gears in the differential or excessive backlash in the gears.
Sorry for bad news. Ask different shops the cost of fixing the problem.