My lamp was making a noise. Now it won't turn on. How to fix?
You need to troubleshoot the problem and this is best done with an electrical portable meter. You probably don't have one, a local harware store would have it. Safety first in everything you do here. If you have fuses remove it from the bathroom circuit at the main panel.
The "lamps" in the bathroom need to be tested in another location and the wiring coming into the electrical boxes need to be inspected (this includes the switch box) and I recommend a new light fixture and/or a new light switch hardwired in the bathroom. Then replace the bathroom fuse at the main panel and flip the light switch on. Good luck.