The IRS and state do not send notification emails when they make a deposit of your refunds. Have you checked the status of it on the IRS's and your state's websites to see if they have given you deposit dates?
If you have successfully filed and your return status shows as accepted in TurboTax, you will need to check their websites to track the status of your refund. They will provide the most up to date status for you there. The date on the TurboTax tracker is only an estimated date; not the exact date. Only the iRS and your state can provide you with an exact date.