Welcome. My name is ***** ***** would be glad to assist.
Can you lower the light kit to expose where the lights 2 wires connect to the fan bottom and see if you see a small square or rectangular item that the light wires originate from in the bottom.
The newer fans have what is called " wattage limiter", which allows only a certain size bulb to be sued in each socket.
This is required by the government to limit over lamping.
They work, but the downside is, they are extremely prone to failure and cause as many problems.
Very hard to find replacements as well, but can provide an online source that usually has the parts needed.
CLICK HERE to see them
Scroll down past halfway on the page and see the "Limiters" section, those are wattage limiters.
Yours may look like one of them.
Try to remove all the bulbs except for one low wattage bulb, sometimes they will start back with a small bulb. Just for a test, as if it is the limiter, it will fail again anyway.
If you do not see anything in the bottom where the light kit wires attach, then your issue may be the in the receiver on the fan that has a relay to switch the lights on.
That can be tested if the receiver is in the fan canopy, it will be the blue wire out of the fan connecting to the receiver.
Test the connection for voltage there to see if the relay is bad
Let me know any questions, etc.