Where can I see my estimated refund date for my state taxes?

Once your return is filed, TurboTax has no control over the processing of your return or the release of your refund.

Here are a couple FAQs to track your refund:
When will I get my federal tax refund?
How do I track my state refund?
Why does Where's My Refund say "approved" but TurboTax still says "accepted"?

The date posted for a federal tax refund on the TurboTax Tax Home website is an estimate only based on the IRS target of 21 days, it is not a promise or a guarantee.  There is no way to estimate when a state tax refund will be issued from the 43 separate states that have a personal income tax.

Once a tax return has been Accepted by the IRS or a State, TurboTax receives no further information concerning the tax return or the status of any tax refund.  

If accepted by the state use this TurboTax support FAQ to check the state tax refund status - https://ttlc.intuit.com/questions/1899433-how-do-i-track-my-state-refund