The RH headlight and marker light are grounded at the same place; a little clip on the ledge behind the RH horn. With that clip not grounded and just the markers on, the RH marker fixture gets grounded through the headlight filaments and the ground on the LH side. Probably some current flows through the turn filaments and to ground at the rear turn signal, too. But not much current (total), so very little voltage drop through the tortuous ground path.
But when you turn on the headlights, the RH headlight tries to draw a lot more current and now there is considerable voltage drop in the ground path. Result, both headlight and marker light are dim (because they aren't getting full voltage).
I expect you can see a similar problem with the RH turn signals, and with the high beam headlights. But nowhere except that clip is the circuit common between the RH headlight and RH marker light, and not common with something else.
Easy way to check, turn on the headlights and check the voltage between that clip and the ground terminal on the voltage regulator. I expect you'll either see several volts; or the headlight will get brighter when you touch the clip.