Tax Year Prior to 2020: Head of household: If you are divorced, both parents each support their own home, and the child spends 50% of the time with each parent, can both parents claim HoH?

My ex and I divorced 2 years ago and we alternate who gets to claim the child as a dependent. Last year was the first year we filed separately and I claimed her as dependent and filed as HoH. This year is my ex's turn to claim the child. 

Now the question: My ex is filing as head of household and claiming our child as a dependent this year. Even though I answered the questions saying "I have court agreement and other parent is claiming the child", turbo tax still  suggests and finds no errors with me filing as HoH. Can I still claim head of household without claiming her as a dependent since I support her at my own house for 50% of the year (she stays with me M-T, with mom W-Th and we rotate weekends)? If both parents claim head of their own households, will that be a problem? If so, what happens? If this is wrong, I assume I'd have to give the money back and pay interest but is there any other fines if I am misunderstanding this rule?

Thanks!