Almost universally, when you're dealing with an insurance company in an injury claim, they want to write only one check to resolve the claim.
That check generally comes at the end of the case.
Rarely if ever will an insurance company agree to pay your medical bills as they are incurred.
Why is this?
Most likely, the insurance company for the at fault party does not want to make it easier for you. Sometimes withholding payment of medical expenses creates stressful situations for claimants;
medical providers want to be paid, and sometimes will put pressure on their patients for payment, even seek formal collection efforts for their bills.
If this pressure results in a claimant deciding to settle earlier in a case, for less money in the case, then insurance company wins.
If insurance companies paid bills along the way, it would take a lot of this pressure off, most likely resulting in
1) higher medical bills and
2) higher injury claim payments coming from the insurance company.
As this never benefits the insurance company, this is very unlikely to occur.