Check the ground and power connections up front, could be the turn signal is finding a ground or power path through the headlight bulb, works fine with the headlights off, but turn them on and the path goes away and the blinker relay "thinks" the bulb is
burned out and blinks faster as a result.

I've seen the same on a rear turn signal on a VW, had a corroded wire inside the lamp socket that made the turn sugnal ground out through parking light, worked fine in the day time, but at night with the lights on and the turn signal on that side no
longer worked.