If you live in a state that collects School District taxes, TurboTax will cover the topic in your state return. Even if you don't owe School District taxes, you'll still need to indicate which district you live in if your state requires a district code. You can look up the name of your Ohio school district here.
While preparing the Ohio return, you'll first be asked for the name of the county in which you reside. Afterward, you'll be prompted to choose from a drop-down of different school districts in your county. (click image below for reference)
The School District taxes are usually e-filed with the state return. To confirm, review the Filing Instructions that are generated when you Download/Print a copy of your return (need help?)