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?
https://www.irs.gov/refunds
If you have a State refund, click on the following link, then click on your State
https://ttlc.intuit.com/community/refund-status/help