As others have explained, the custodial parent can use the child for the EITC even if the noncustodial parent is claiming the child as a dependent and claiming the child tax credit. So TurboTax was correct to allow you to use the child for the EITC. That cannot be what prevented the other parent from claiming the child as a dependent.
It's not clear what the problem is. We cannot see your tax return. It sounds like you might have accidentally claimed one of the older children as a dependent. Look at the list of dependents on the first page of the Form 1040 that you filed. A child who was supposed to be claimed by the other parent should not be listed there, even if you used that child for the EITC. Look at the actual Form 1040, not a screen in TurboTax.
