TurboTax gets no information after your e-file has been accepted. TT will not know if any sort of refund has been issued to you, nor can TT find out for you. You have not mentioned what state you are in. Some school tax returns can be e-filed; others have to be filed by mail. If the school tax return was e-filed, it would have been e-filed when you e-filed your state return. Check the page of filing instructions that prints out with the pdf of your tax return and see what -- if anything-- it says about your school tax return. And check with the school district/school taxing authority itself to inquire with them.
**Disclaimer: Every effort has been made to offer the most correct information possible. The poster disclaims any legal responsibility for the accuracy of the information that is contained in this post.**