You are correct, you are only subject to school district income tax (SDIT) for the time you lived in the taxing school district (SD). But, TurboTax (TT) is not capable of doing that calculation. You have to enter the taxable amount of income manually. After telling TT, in the SD interview, that you were a part year resident, you will be presented a box to adjust your earned income (enter the taxable amount)*.
*If you lived in a traditional SD, instead of an earned income SD, you will be presented a box to enter the income NOT received while a resident of that district (a deduction). TT has to do it two different ways to match the SD tax form (SD 100).