Bad ground on the non-working lamp.
May be as simple as remove and reseating the blub.
WHY.
When running lights OFF the turn signal works because it grounds through the running lights via the common (Shell) connection on the bulb.
With the running lights ON the DIMMER element in the lamp does the same via the turn signal lamps but when both are "Lit" There is no ground.
Start by removal and re-seating of the bulb. Then start chasing fora bad ground on that socket.
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times