For each year, only the person who has physical custody can automatically claim the child. The custodial parent can release the dependent to the non-custodial parent with form 8332, it can be signed for one year at a time, or several years according to a schedule. The custodial parent must mail the signed form 8332 after e-filing the rest of the return, they should keep a copy. The non-custodial parent does not file the form or any extra paperwork.
Be aware of all the rules: while form 8332 can transfer eligibility for "dependent" status (which means the $2000 child tax credit), it does not transfer eligibility to file as head of household, or claim the child care credit or EITC. Those benefits always stay with the custodial parent and can't be waived or transferred. The custodial parent should answer in Turbotax they they have custody more than half the year (pick 7 months or more), there is a custody agreement, they will give the other parent a form 8332. This will set the child's status as "non-dependent, use for for child care credit and HOH". The non-custodial parent should answer that the child lives with them less than half the year, there is a custody agreement, and the other parent has given them a form 8332. The non-custodial parent will get the dependent tax credit but will not be eligible for EITC or head of household status (unless they have other qualifying persons in their home). Turbotax will also tell them after they file, to print a form 8453, which is a special cover page that they use to mail the original signed form 8332 to the IRS after e-filing their return.
