TurboTax cannot give you a direct deposit date. Only the IRS & State can do that.
Once your returns have been accepted , only the IRS (& State) would know when your refund will be issued. States differ when they issue refunds from the IRS.
You can checked the IRS website Where's My Refund for your information?
If you have a State refund, click on the following link, then click on your State