First as I understand from the thread there are two switches that independently (not 3-way) control 2 different light fixtures in your kitchen.

The diagnosis process is just a matter of elimination and you can do it methodically from end to end or by the most likely failure I am going to mix those up and relay the way I would do it.

If both of those fixtures are in the kitchen then they are almost certainly on the same circuit BUT I would look at your load center to make sure a breaker hasn't tripped. If it is reset and you will either be good or if it pops again you have a different issue with a different set of more complex diagnoses.

Then I would take a bulb that is know is good (from the other fixture maybe or new after testing it in the other fixture) and put it in the faulty fixture. If it doesn't work I would likely put all 4 good bulbs in the fixture. I have never seen a residential light fixture wired in series but you see some really odd stuff in house wiring.

If the new bulbs didn't work I would take the fixture apart (power off) and look for loose wire and tighten the wire nuts.

If that doesn't fix the issue I would test to make sure power is coming in through the NM (ROMEX) cable to the fixture wires (likely lighter gauge and stranded). If you have power there with the switch on then there is a connection issue in the fixture (redo all the wire nuts) or something in the fixture itself is bad which you could also track down by tracing the wiring. I would pull the whole thing down and use a battery and multimeter to determine where the fault was.

If you do not have power to the fixture you need to test if you have power coming out of the switch when it is on. If you do then you have an inwall issue that is going to have to be tracked down. Maybe a junction box that shouldn't be there with loose connections, squirrels in your attack, or rats in the wall kinda thing. This assumes there is not another device on the circuit but I would expect there is only the one light fixture.

If you don't have power coming out of the switch then you need to pull the switch out and look for loose wires or loose connections. If tightening any loose connections or replacing any loose wires (you gotta know where that loose wire goes) doesn't fix the issue you need to test if you have power coming into the switch itself. If you don't and you checked all the breakers then you have a potentially complex inwall/ceiling issue in the run back to the load center, it could be the cable but (more likely) it is on a previous device (switch receptacle). If that switch is in a multi-gang box with other switches they most likely are all on the same circuit so if one works it is likely downstream from there.

I didn't explain how to test for power in the various places on an energized circuit because I am not going to help you potentially hurt yourself but if Google explains it to you then it is on you an Google.

Wiring issues like this are simply testing step by step through the electrical path to find where the fault is. It is actually incredibly simple but it requires testing energized wires (unless you have some testing equipment beyond just a multimeter). Just go one step at a time through the electrical path and rule things out until you find the issue.

I hope this makes sense, it is something I could do faster than I can explain it by typing and sometimes things that are just intuitive are the hardest things for me to explain.