OBGYN won’t bill my health insurance until after baby is due….is this right??

I’m currently 6 months pregnant and live in Texas. I’m going to Fenóm Women’s Care in Fort Worth for my prenatal care visits, everything has been going smoothly so far but at the end of one of my last visits I received a bill for $2730. This bill encompasses all OB visits, delivery with the doctor and post care visits only. All lab work, hospital charges, and additional sonograms are separate. They are stating that this is due in its entirety by September 9. This is my second pregnancy and I’ve never gotten a bill from an OBGYN like this and from my conversations with other women, this is not something that is common. When I spoke to my insurance company they stated that I should have never received a bill like that because everything should be covered within my insurance. They stated that they haven’t received a medical bill from them since my first visit in April and that may be why I received a bill like that, but otherwise they have no idea why I would be receiving a document like this. I spoke to the staff at my OBGYN office and they stated that the do not bill insurance until after the baby has been delivered. This seems really off to me and doesn’t feel right. Is this normal?

I just wanted to know if anybody else has had an experience like this? My pregnancy has been really easy so far, but this bill has completely stressed me out to the point of tears because we don’t have this kind of money and the bill is due its entirety by September 9. We’ve been trying to set aside as much money as we could to try and pay it off, but that has left us with nothing to prepare for the upcoming baby. We’re due on December 2nd and we haven’t been able to purchase anything to prepare for the baby since we’re putting all of our money away for this bill. Any advise would be greatly appreciated!!