If two parents are living together unmarried and share a child, one parent lists the child as a dependent, answers "no" to the question about a custody agreement (because that only applies to a court order between parents who are divorced or separated) and the other parent does not list the child on their return at all. Parents living together can't "split" the tax benefits of a dependent. You can't claim any day care expenses, but you paid expenses for care so the other parent could work, the other parent could include those expenses as if you gave the other parent the money and the other parent paid the caregiver.
For cases where parents are divorced or separated or living apart and sharing custody, only the parent where the child lived more than half the nights of the year can claim the childcare credit, along with head of household status and eligibility for EIC. If that is not you, you can't take the credit for your childcare payments.
If you are the parent with custody more than half the year, then the other parent can only claim the child if you give them a signed form 8332 dependent release. In turbotax, you would indicate the child lived with you more than half the year (choose "7 months" or more) and there is a custody agreement and you are giving the other parent a signed form 8332. The child will not be your dependent and will not qualify for the $2000 child tax credit or the stimulus rebate, but is still a qualifying person for head of household, EIC and the childcare credit.
