In case of dispute, it is right to go to a public hospital, but it is wrong to transfer to a private hospital.
When two cars collide, both sides may be at fault, but you are behind, and the main responsibility lies with you. The other party is injured, regardless of the size, medical expenses, hospitalization expenses, lost time, nutrition expenses, car repair expenses, etc. You should pay for it. It is suggested to negotiate directly with the injured, be sympathetic and sincere, and make appropriate compensation as soon as possible.
You can't just give money, you need surgical records, hospitalization documents, etc. Keep them for the future. Payment invoices should be properly kept. If there are arbitrary charges, you can complain to the medical and industrial and commercial departments.