Deposit for giving birth?

I live in Brooklyn with my wife and 4 year old. We have a family health insurance plan with Aetna (H2) that costs me over $2000 a month. We are expecting our 2nd kid and my wife is going to Mt. Siani in Downtown Brooklyn for her appointments. All the doctors are in our network. The last 2 times she went for checkups they said she had to pay a “deposit” for giving birth and they wouldn’t see her for her appointment until she paid the money. The 2 payments totaled $4,182.37. I find it really strange that we would be paying that much money directly to the hospital and that they would require it before seeing her. When she gave birth to our first child we went to a different hospital that never did anything like this. Is this normal? Can I get this money back? Should I cancel the credit card payment? I’ve never had to deal with a medical claim before. I am also worried that if we go to the hospital for any sort of emergency regarding the pregnancy they could just ask for any random amount of money before seeing her even though we are fully insured.