Is this car a newly wired car and you are checking out the lites for the first time?
Since the right rear works without the headlite switch on indicates that the ground is GOOD.
Typically the headlite and taillite circuits are separate from the brake and turn.
I would suspect the switch or wiring . It could be even something as simple as the switch grounding out the right taillite wire when pushed on or causing that wire to be disconnected when the switch is on.
Check how it works with the switch out and dangling. Look for loose wires or bad stacon connections.
Stacon connectors should be crimped twice. Once on the wire and than also crimped to the insulation.That will help keep the wire from flexing and breaking where it enters the stacon.