When headlights are turned on they dont come on for 5 min when its cold?

"Delayed" Headlight Turn-on

There could be many causes for this issue, and you don't say what make your vehicle is, so I'll provide my experience with this problem.

I have a 1993 Ford F-150 pickup which developed the same problem. After all my hours of troubleshooting over several weeks it turned out to be that the main headlight switch had become defective.

I replaced the headlight switch which solved the problem, but it was expensive!!!!!

I, as I always do with any replaced part, dissembled the old switch to study it and determine the cause for failure.

What I found [my analysis and opinion] was that Ford had saved money by using a switch in which the Copper [or Brass] contacts were made of material that was too thin ["lightweight"] for the job it had to do. Thus the electrical load of the lights using the switch caused the inadequate contacts to heat up hot enough to cause [over time] a buildup of CORROSION which made the heating problem even worse, resulting in the contacts finally "cooking out" until they were ruined.

Before buying and going to the trouble to install an expensive switch, I suggest you have a qualified Professional Automotive Electrical Technician troubleshoot the headlight electrical circuit(s) to make sure an expensive, "good" part is not being replaced unnecessarily.