Are you sure the turn signals are the bright ones though? Since the indicator lamps being on with the headlights won't stop the turn signals from flashing if they're simply reversed, you'll still get the lights flashing.
But you'll in fact be getting the bright filament lit with the running lights, and the dim filament flashing with the turn signals. Hard to tell the difference unless you're checking real close.
When you turn the signals on, do either of the turn signal indicator lights flash also? Or do they just stay solidly illuminated while the turn signals do their thing?
Paul