First time filing having a child. TurboTax asked for the SSN so I provided it. I put my partner was going to claim the child, somehow she cannot claim and neither can I .

Sounds like the two of you were confused and tried to enter information about your child on both of your tax returns.  If all of you are living together as a family, then only ONE of you should enter any information at all about the child on a tax return.   Unmarried couples who live together with their child cannot "share" the child as a dependent or share the child-related credits on a tax return.  ONE of you can enter the child's information and get all of the child-related credits for which you are eligible.    The other parent simply files as Single and makes no mention of the child on their tax return.

If you have determined that your partner should claim the child, then they have to file by mail instead of e-filing.  Your use of the child's SSN on a tax return has blocked anyone else from e-filing a return using that SSN.  Amending your return will not help her to e-file.  The problem will exist for this year.   You will eventually need to amend your own return and remove all mention of the child from your own return.