Adalbert, does your car have its tail lights on all the time, that is, does your daytime lighting set up include the tails lights?
It sound as if the ground connection you are using is in fact not ground at all but something else.
Does your car use a 2nd filament in either the brake light bulb or the tail light bulb for turn signal or a separate bulb. Clue is that if your rear turn signals are amber then it is a separate bulb.
if you are splicing in then comes a 2nd question, on the tail/brake light assembly in the car is theer a common ground wire and a "signal" wire to each bulb or a separate pair of wires to each bulb.
It would be good to check if,with the trailer attached,that ALL the car lights work correctly and at the correct brightness.