2003 Grand Marquis
 

While driving at night when I turn on the headlights after a few minutes they go off. Then after turning off the car and starting it up again the come back on

Hey yoda!

Hmm. This is just a thought, but it sounds like maybe your timer for the headlights may be involved? Perhaps something in there is being told that your car is off, and automatically turning them off with the delay timer.

Food for thought!