Sounds like a broken wire more likely than a fuse.
There are fuses for taillights, left and right turn signal and brake lights. But if the RV lights work okay when not plugged, those fuses on the RV have to be okay!
As a seperate issue, if the lights on the car work when unplugged, the fuses on the car are okay.
What seems to NOT work is transferring that power from the RV lights, signals, etc working to the car.
Hard to imagine that several wires, each feeding power to the car for different lights, would all go bad at once without you spotting a major problem.
But the one common item which migh knock all out is the ground.
There should be a single point in the trailer plug on the RV which passes ground from the RV to the tehter and to the car wiring.
There is a wiring standard which we hope folks followed as shown:
If you look into the connector at the RV rear, the pins should look like this.
A meter would be best but if you have a test light or some way to try things, you should be able to tell if things are correct!
When you turn the RV headlights on, there should be 12VDC power on pin 3.
If you touch pin 3 with one side of a test light and then pin 1, which should be ground, the light should come on.
That proves you have good battery as well as good ground to that point.
This is to prove that power is getting to those pins on the connector, so to chase it further, you can connect the tether cord and go to the other end of it to see if the battery and ground get to the car end of the tether.
As you move closer toward the car taillights and turn signals, there is likely to be some point where the battery or ground is not getting there and look for a loose wire?
CAREFUL! Don't let some metal item slip and touch both the ground pin and a hot pin at the same time. It will not hurt you but it WILL very likely blow a fuse on the rV and that is not progress as you then have to hunt that fuse!