You cannot claim the children on your return unless the other parent signs a form 8332 that allows you to get the $500 credit for a dependent and child tax credit. If your ex does not sign such a form you do not put those children on your return at all. Even with the signed 8332, the custodial parent is the only one who can get earned income credit or childcare credit. Without a signed 8332, the custodial parent gets all of the child-related credits.
The non-custodial parent, if allowed by divorce decree or consent of the custodial parent on form 8332 or similar signed statement, can claim the child as a dependent and qualify for child tax credit if the child is under age 17 or family credit if the child is older. For post-2008 divorce decrees or agreements, form 8332 or similar signed statement is required. The child tax benefits cannot be split any other way.
It depends on how you answer all the questions in TurboTax as you go through entering the child how each of these is handled.