TurboTax never pays your tax. If you first tried to e-file you might have entered bank information to pay by direct debit, which means that the IRS (not TurboTax) takes the payment from your bank account. But when you file by mail you cannot pay by direct debit. You have to either include a check with your mailed tax return, or pay online on the IRS web site using Direct Pay or one of the other options offered. Check your state's web site for how to pay your state tax.
Check the instruction cover sheet and see how you were suppose to pay it. Like by check, etc. Did it print out a 1040 V voucher slip? Look at your return and see if the bank info is printed on it at the bottom or just xxxx's. When did you mail it? Maybe the IRS hasn't processed it yet. If they had they would have sent you a bill for the tax due. Did you mail federal and state in separate envelopes to different places? If you mailed them together then one wasn't processed at all.