The general rule is that the IRS issues federal refunds within 21 days of the return being accepted; sometimes it can take much longer, for various reasons. And you can see reasons here Why is my IRS refund taking longer than 21 days? And state refunds vary since processing times vary from state to state.
If you have successfully filed and your return status shows as accepted in TurboTax, you will need to check the IRS's and/or your state's website to track the status of your refund. They will provide the most up to date status for you there.
They do not send any updates to TurboTax after they accept your return. Once they process the return and approve your refund, they will update the status with a deposit (or check mailed) date for you, on their website.
You can use the IRS Where's My Refund site for the latest status on your federal tax refund.
And you can use your State Department of Revenue's website to check status of state tax refund. Choose your state here How do I track my state refund? and it will give you steps to track the status.