Does the car have any other relay or fusses in other place?

Have you checked under the dashboard yet? Today's cars and trucks have two fuse boxes. The first is on the driver's side of the vehicle, usually under the kick flap to the left of the brake pedal. There should be a pop-off cover or box located on the firewall that contains a number of fuses. It isn't likely to be a fuse fault, though.

It sounds like you are experiencing problems with a thermocouple style switch. In other words, when the the switch or device reaches a certain temperature, it switches off.

To find this particular device, you will have to obtain a service manual for the electrical and fan systems and then you will have to find the diagrams that cover the fan. Study them and you will likely find that there is a special symbol there that shows there is a heat-activated or thermocouple switch. If you identify it and feel comfortable handling the job, remove the old switch and swap in a new one. The chances are good that this is the whole problem. I honestly doubt it was a fuse in the first place.

One other suggestion is this, look to see if there might not be a cooling system switch that handles this function. You'll need the service manual for that, too.