2001 toyota solara blower motor shuts off when headlights ara turned on

check ground(s) usually coming out of the harness trace back from headlight plug (on problem side ) until you find a ground coming out of harness and bolted to sheet metal in engine bay make sure there are no loose,corroded, broken ,or burnt wires

Check blower motor switch . When wires burn they short out the next thing in line and relays under hood

fuse problem or check the headlight bulbs double check wiring connections to.

check wire's and earth on the unit, if a faulty earth this will cause lights to do this,

I would replace the socket for that light that is shorted.

The actual switch may be broke, even though it is turned off. It may be stuck in on.

No.

Probably not. If it runs on high, it is probably good.
Unless is sounds slow or 'tight' on high, or squeaks or chips , etc. Run the fan with the engine off and then shut the fan off and listen and see if you can hear how long it takes to slow down. If it stops in just a second or two, the bearings are tight (dry). If it spins down normally, maybe three to eight seconds or more, the motor is fine. You really don't need this check.

It is most likely your blower relay.
Most cars have one or two relays to control the fan motor, which is a single speed motor that the relays and speed switch and blower resistor control the power to.

One relay is for high speed, also switch contact.
One relay is for other speeds and divert current through a blower resistor located by the fan duct under the dash or under the hiid by the fan.
Also the switch contacts for each speed.

It cad be the dash switch, which has to handle the blower resistor current for all the lower speeds is burned out at those speeds.
It can be the low speed blower relay, as I mention above, is burned out, this is most frequent cause.
it can be the blower resistor is burned out, this can give no low speeds, or loose speeds below a certain speed setting.

Try you manual for the blower relays' positions. Try swapping with the horn relay to test if it is the same type of relay.
Take out the blower resistor carefully (pretend you are defusing a bomb or playing "operation" game as it has exposed coils that are delicate and if you bump the sides of the opening as you extract it you will damage it and short the coils against each other). the wire coils are very fine bare wire coils that look like springs and if touching can burn out.
Switch is harder to test because hard to get at, check the other things first.
You should be able to test for voltages at the resistor plug to see if power is coming from the switch, but wiring schemes can vary.
Post back or use shop manual to troubleshoot the switch.

Yes could be the blower relay or loose ground in the blower circuit.
Try your local library they may have a manual with the cicuit diagram.

You need to replace the heat control reostat or relay. On Toyotas' it is behind the dash nest to the blower motor. This unit is held in place with two small screws and has wires and a plug on it. Pull it out it may look like it may have coils that go between

Basically, you have a fan relay that has shorted and burned. The relay is used because of the different power setting needed for the fan speeds.

Trouble shoot all fuses, incase you blew a fuse when the relay went. The relay burned but that did not effect the motor, thats why the motor runs well.

To find the location of the hi speed fan relay, check with your local parts store to see if they can bring it up on a computer.

Bill