Have a 2004 Dodge Sprinter.Headlights not working unless turning on the bright lights and also left blinkers are on. At this point if bright lights are turned off then the headlights working. Creepy
Check the ground connections. It sounds like corrosion is breaking the circuit until the higher current of high beams and turn signal restores the connection.
Along with cleaning suspected faulty ground connections check the light switch if the trouble is still there after cleaning the grounds. By making a fairly long jumper lead with clips on each end you can test for bad grounds by tying one end of the jumper wire to a known good ground point, like the minus side of the battery. Then connect the other end of the jumper to a suspected bad ground wire connection.
Does this have Day time Running lights and do they work? I would start with the ground. Also, a bad Multi-function switch would be a possibility.