I wonder if there is a bad ground for the front light unit on the right hand side. When the headlights are on high beam, enough current flows to overcome the resistance in the bad ground. My only reason for making this guess is that years ago, we had a problem in our house and half the house lost power. All the fuses were good. However, we could restore power by turning on the clothes dryer. It turned out that there was a loose connection in the electric meter for one leg of the line coming into the house. Turning on the 240 volt clothes dryer drew enough current to overcome the resistance in the loose connection at the meter.
Unfortunately, this doesn’t explain why the emergency flashers work on both sides. My second guess is that the bulb in the front is wrong–possibly a single filament bulb where a double filament is required. You might try putting in new bulbs on the right side, making sure that the bulbs are the type called for in the owner’s manual.