TurboTax gets no information from the IRS or the state after you file your tax return. TT cannot check on it for you.
Federal and state refunds come from completely separate entities, There is no rule as to which one will come first or how much time there will be between their arrivals.
Make sure your state return was accepted: https://turbotax.intuit.com/tax-tools/efile-status-lookup/
To track your state refund: