TurboTax does not pay anything in terms of estimated tax payments; you need to do that yourself either by sending in the 1040-ES vouchers or making an online payment (see link below).
https://www.irs.gov/payments
Short-term capital gains are taxed at regular income tax rates. Since I do not know your total income (and the type), I cannot determine exactly how much you would owe to the state or feds (if anything).
