CygnusX1 has an on-line TSRM. It's for a 1990 model, but it works.
Actually, I've found that 9 times out of 10, in the Supra, it's the passenger side switch that's at fault.
The wiring is done so that it requires that neutral contact (or agreeing contact) must happen in both switches so the the window motor will operate. If the switches are reversed, the window motor will do nothing. This is a more or less normal wiring setup for power windows.
The driver's side switch is a sealed unit. The passenger side one can be disassembled. There are 2 copper bars inside the switch, and contact plates for them. These parts can be (And probably need to be) disassembled and cleaned. Clean the contact plates while you're at it. Use some electrical contact grease on the parts afterwards, or (like me) you'll find that your fix doesn't last for too long before you need to do it again.