Headlight Wiring Problem

Your described symptoms sound (read) like a classic case of a broken or accidently disconnected ground wire that is common to both the headlight and parking/turn light.
Normally, with the headlights on, power flows from the battery to the low-beam flimant, and then to ground. When the ground is missing, the current finds a path through the high-beam filament, which is why the indicator is on (perhaps dimmer than usual). The raised gound wire voltage appears as a missing or open turn signal bulb to the flasher, which flashes at a faster rate as a diagnostic.

Perhaps someone has a shop manual for you model year that shows where the left headlight ground lug is supposed to be connected.