My water well turns on and off continuously every two or three seconds when water is in use. Is this normal?

No. Sounds like a water logged tank. Is the tank a bladder type?

If it turns off and the gage falls until it turns back on I'd suspect a faulty foot/check valve. I don't think it's the bladder tank. If it were the pump wouldn't cycle but would turn on every time you made a draw. Good luck, nTom

when water is in use

I've got to vote tank.

Hil Mmannix,
I missed the part where the only happens when you made a draw. I agree with the experts. Sure sounds like a ruptured bladder tank to me. Replace the tank and you be just fine. Good luck, Tom