Indeed strange. Hi beam lamp uses same chassis ground path as illumination, and turn. Hi beam shouldn't come on until it gets the same power as the headlights on high beam.
Turn should work with switch on engine off but only if there's enough power supplied from battery and wiring. Its possible that your illumination actually does work just so dim you can't see it. We've seen this condition posted before. Are you working outside in daylight?
You could add a ground wire at the headlight switch. On the inst' housing too. Proves ground was or isn't the problem.