Ohio does a convoluted tax calculation for non-residents/part year residents. It calculates tax on total income, then it calculates a non resident/part year resident credit, which it subtracts from the tax it calculated on the total income. The credit is calculated as your non-Ohio income divided by Total adjusted Income multiplied by the total tax. TurboTax (TT) does this by allocating your income as either Ohio or non-Ohio. W-2 income will be allocated by the state name abbreviation shown in box 15 of your W-2. TT will ask you, item by item, in the state section, how much of your other income is Ohio or non-Ohio income. Make sure that your non-Ohio wages show XX (Other state postal abbreviation) in box 15 of your W-2 screen, with the XX amount in box 16.
Also be advised that you may have to use a work around to get TT to prepare the return. At the w-2 screen split the boxes 15-20 W-2 info into 2 lines. On the 1st line leave OH in box 15 but change box 16 to the correct amount; leave the OH withholding in box 17. On the 2nd line put your other state abbreviation in box 15 and the correct state wages amount in box 16; leave box 17 as is..