As TRUCKBIZ said. It's the ground. Probably the ground of the right front bulb socket/housing/body attachment. Run a test wire from the battery ground post to various metal spots in and around the lamp while it's set on blink. Start with touching directly on the bulb case. If that makes it all work, then find out where it is loosing ground from bulb case to body/battery.
Bulb case: The brass part of the bulb that the glass is attached to. Take off the lens to get access to it with a test wire probe.
Repair: (if testing confirms ground issue at bulb)
1. Examine attachment method of the lamp components.
2. Look for looseness, corrosion, rust. Clean/burnish or replace bulb socket and housing.
3. Compare to other side to see if anything is missing/damaged.
4. Be sure bulb bayonet pins have good contact to socket.
5. Add a ground wire.
If you are saying that both front signals flash when just the right is activated, that means the right is looking for ground and may be feeding back thru the other bulb. It's weird, but that's what it does sometimes.
Of course, there is the possibility that you have made a wiring error.
You never said anything about what the rear lamps are doing.
Does the right rear signal working properly?
Does the left front and rear work properly?
The ONLY issue is that when right is activated, both front lamps blink and......... when hazard is activated, right is dim. No issue with rear. Correct?
6V or 12V?
