Here you will see two hospital childbirth delivery videos. These both show a fairly typical hospital birth experience, including epidural anesthesia and coached pushing with the mother laying on her back. Also fairly typical, you will notice the babies having their umbilical cords clamped immediately and taken from the mother for medical assistance.
I am still looking for a good video of a natural birth in a hospital to add. Though not as frequent, they do happen- I know, I had two of them!