When a burn turns white you usually have a second degree burn.
A second degree burn is a serious enough burn that you should go to the emergency room for treatment.
The second degree burn that turns white is more serious because the damage extends beyond the top layer of skin.
If the burn is healing though then you may not need to go to the emergency room or urgent care for treatment but at least have a doctor look at it.
The second degree burn should heal fully though within a few weeks to a month or two but you will likely have burn scars left over.