You do not report rent you paid on your federal return. There is no deduction or credit for rent paid. However, if you claim a deduction for 'Home Office' use, a portion of your rent may be included in that calculation, but this would be the only place. The rentals and royalties section is for Rental Income you received, not for rent you paid.
Additionally, if your state has a renter's credit (some do, some don't), TurboTax will ask for that information in the deductions section when you are ready to file your state return.