The key question is where did the child live more than half the nights of the year, with you or the other parent. It sounds like with you. In that case you can claimed head of household.
If you have an agreement or divorce order that says the other parent can claim the dependent in certain years, you need to give the other parent a signed form 8332 (dependent release) for that year. You can download it from the IRS web site or access it in turbotax under "Other tax situations." Using the form 8332, the other parent can claim the dependent exemption and the child tax credit. But the qualification for EIC, HOH and dependent care credit always stay with the custodial parent and can't be transferred. In Turbotax when you do the dependent interview, say you had custody more than half the year (7 months or more) and then say you are releasing the dependent exemption to the other parent with a form 8332. Turbotax will assign a status "Non-dependent; use for EIC and HOH".
The other parent gets the dependent exemption and the child tax credit but can't claim EIC, HOH and dependent care credit unless they lie and say they had custody more than half the year. This is important: the other parent can't claim head of household because they would have to say that the child lived with them more than half the year. The parent with less than 50% custody only gets a partial dependent benefit, that's the way the rules work. See IRS publication 501, pay special attention to the special rules for children of divorced or separated parents.
Read IRS publication 501. https://www.irs.gov/uac/About-Publication-501
In the event the other parent does claim that the child lived with them, you may be blocked from e-filing. Just print your return and mail it in, the IRS will investigate the overlapping claim. You may need proof that the child lived with you more than half the year such as school records, doctors records, or other proof of the child's address.
