Personal injury due to car accident that was the other drivers fault; should I be paying the co-pay out of my pocket or should my car insurance be paying it?

Your attorney should not be giving you any money as a loan against a possible future recovery.  The contingent fee agreement that you almost certainly signed here is the most that an attorney can do for you, the way most lawyers view their ethical obligations.  For example, suppose the loan was $10,000, and the defendant offered a $25,000 settlement; settlement offers are almost always somewhere in between the best and worst possible trial results.  If the attorney was worried about getting less than $15,000, he'd want to settle, because that would be in his best interest, even if it wasn't in yours, and this is why it usually isn't done.

Keep receipts and records, for whatever you do have to pay out-of-pocket here.  Your attorney will need that, to prove your damages.