The only parent who has the automatic right to claim a child as a dependent is the parent where the child slept more than half the nights of the year. This parent is entitled to the child tax credit, and also to file a head of household, claim EIC, and the dependent care credit. The other parent is entitled to nothing. The IRS doesn't follow agreements, and there is no such thing as 50/50 custody. You have to count the number of nights, and most years have an odd number of nights so one parent has more than half.
If the custodial parent is supposed to give the dependent to the non-custodial parent in certain years, that is done by giving the non-custodial parent a signed form 8332 dependent release. This allows the non-custodial parent to claim the child tax credit only. The ability to qualify for EIC, head of household, and the dependent care credit, always stays with the custodial parent and can't waived, shared, or transferred.
If you are the parent with custody more than half the number of nights, then in Turbotax you say that the child lived with you more than half the year; that you have a custody agreement with the other parent; and you are giving the other parent a signed form 8332. Turbotax will properly allow the child to qualify you for EIC and head of household.
Your ex will say they had custody less than half the year; there is a custody agreement, and the other parent did give them a signed form 8332. This will give them the child tax credit only. They will then be required to mail the original signed 8332 form to the IRS after e-filing (keep a copy of course).
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Your ex probably tried to claim the child without saying he had the release form, as if the child lived with him more than half the year, entitling him to more benefits. This will create an e-file conflict. He probably needs to delete the child dependent and add him back properly, and you need to give your ex form 8332.
