Hello, sounds like you have a bad earth to the front lighting. This is quite common and causes some lights not to work and some for power to back feed through other lights, it is most common on the rear clusters and results in the lights operating in a unusual fashion, for example flashing through brake lights when indicating.
With the lights on and bulbs removed, one pin should be positive, as you have found, the other should be negative.
Depending on what equipment you have, we could look up the details for your car (not yet provided) and see if we can find the wiring diagrams.
Alternatively, as it seems you have the positive, hence the fuses appear okay, I would consider running another negative wire to the negative side of the lights.
This could be taken back to the battery, or fixed to a nearby clean piece of metal that is 'earthed' to the negative side of the battery.
Let me know if you want any further help.