They tried to make a law once called the "Tender Years" doctrine. It stated that an infant must be in the custody of the mother.
It got shot down as unconstitutional. There is no such law on the books.
Now, I DO understand what you are saying here. Even though the LAW says that both parents are equal before the law, it just isn't that way in practice.
I was just pointing out that from a technical standpoint, the law says that both parents are equal in status as parents and neither has official custody until a judge says so.