You cannot claim a dependent for only half a year.
Only the person with whom the child stayed with more than 50% of the nights gets to claim him/her. So you'll have to count up the exact nights.
There is also an option the the custodial parent (assuming you do not live together) can give Form 8332 to the non-custodial parent to allow the non-custodial parent to claim the child's (1) exemption and (2) Child Tax Credit. If the custodial parent qualifies, the custodial parent would still claim (3) Head of Household, (4) Earned Income Credit, and (5) Child and Dependent Care Credit (daycare).
If you do that, just follow the prompts in TurboTax, and make sure the non-custodial parent mails Form 8332 with the tax return.
If you are referring to Form 8332, TurboTax should have the option to e-file your tax return, BUT Form 8332 would still need to be mailed along with the special Form 8453.