After you file

Once the tax return has been accepted by the IRS then TurboTax does not receive any further information concerning the tax return or tax refund.  You have to check the status of the tax refund on the IRS website.
This is the same with any other tax preparation software you use.  Unless you use a paid tax professional to prepare your tax return then they can check the status of your tax return and contact the IRS on your behalf.