It is not compulsory to carry a first aid kit in the car. Also, if you do want to carry one, it does not need to be very large, or carry everything. In the first instance, in the event of a serious accident, the most important thing is to have materials available to staunch bleeding for as long as necessary until specialized help arrives.
So the must-haves in a car first aid kit are gauze, dressings to stop bleeding, and bandages. To complement these elements, it is good to also carry scissors, tape and latex gloves (and make sure everything is sterile, to avoid infections).