A lot of modern vehicles do not like the trailer wiring
to be grounded to the car chasis. Also many vehicles will advise to get a power booster to run the additional load of the trailer lights.
Start with the documentation
of the vehicle. There should be a section in the owner manual about trailering.
Then troubleshoot the trailer lighting
separately. Trailer lights are set up in many various ways. Primarily you will see one running light circuit that lights up the side lights and one filament of the (usually) two filament tail light bulbs. On occasion you can have dual bulbs in the tail light.
The second circuit is the tail light "bright." This is where things are usually broken down. The bright filament needs to be powered by either the turn signal, separately and intermittently, or the brake light power which is both filaments. So going to the vehicle should be 4 wires minimum. Ground, running, left high and right high.
Ohm out each circuit and as stated check all the grounds. When I wire a trailer there are no grounds to the trailer. Each circuit is a twin wire, tied back at the front and grounded through the vehicle through the plug
. The fewer grounds and connections the fewer failure points.
The final piece is to check the mapping of your 6 to 4 adapter. Using the DC function on your meter, map out the outputs from the 6 prong on your vehicle, then plug in the adapter and make sure the output (4-prong) matches what the trailer needs.