Headlight switch's have a built in circuit breaker that will cause this if there is too much draw or the bi metal circuit breaker is just old/weak. Most old headlight switches can be disassembled and checked/cleaned etc. You can bypass the internal circuit breaker and install one inline if you cant find another switch. Headlights use a circuit breaker rather than a fuse for safety reasons.

Thanks
Randy

I had a 65 Galaxie that did the same thing. It was the breaker tripping in the headlight switch.

If you cant find or fix the switch you could always put a external relay in so the amperage is not going throug the headlight switch, you just have to run the headlight feed wire to the relay ( may only draw 1 amp ) then run fused wire from bat or fuse panel to relay and out to headlights. If staying 100% stock not the way to go, but if all you want is headlights to stay on, works great.