Yes it is absolutely normal practise, however....while it is probably too late to arrange this now, some vets and insurers will agree to have the client pay the excess direct to the vet and then for the insurance company to pay the vet directly for the rest once the treatment is finished.
Not all insurers will do this, nor will all vets agree as they are then having to wait for their money (which affects their cash flow) and they are at some risk is there turns out to be any dispute over the insurance i.e condition excluded, pre-existing condition, insurance lapsed etc.
The other way you can do it with some companies is to put in a claim AFTER each "section" and then do a "continuation claim" for the next section after that is paid e.g you claim for the part for the first vet, then you claim for the referral vet's bill, then you claim for the follow up treatmnet from the original vet, if that occurs. Being a continuation claim you do not pay another excess each time, unlike a new claim.
You still have to wait for the turnaround time of the insurance company processing each bit of the claim, which varies greatly beween companies, but you don't need to wait for the end of the entire course of treatment before you can reclaim anything.
However some practices charge a fee for the time it takes them to do the claim form (I know one big veterinary referral practise that has two people employed full time just to do insurance claims- it can be very time consuming especially after dealing with queries etc) and if the vet you use does charge for this (which you cannot reclaim from the insurers) , the more separate forms you put in during the course of one treatmetn, the more it will cost you in unrecoverable claim form fees, so swings and roundabouts.