The main determinate is what state. There are over 40 state tax agencies, each doing its own thing. Some states are fairly quick with refunds, while other states (such as WV, WI, NC, GA, LA, AL, and others) can take a long time processing returns--even several weeks--as they crack down on detecting fraudulent returns. Plus state tax agencies have likely been affected by the Coronavirus situation creating additional delays. You may be able to get an idea about how long it takes in your state from the FAQs at your state's tax agency website as I will mention below.
Of course, you first need to be sure it was successfully filed. Here's how:
If you efiled, be sure your efiled state return was accepted. If you used Online TurboTax, you can check by looking in your Online account at the Tax Home.
Or did you choose to file by mail instead? TurboTax does not mail it for you. If you chose that method, you have to print, sign, date, and mail it.
The FAQ below tells how to check on a state refund and has a link for your state.
TIP: While you're at your state's tax website, also look for a statement or FAQ that may say about how long it takes to get a refund in your state and how to long to wait before phoning the state tax agency. Many states have such information on their websites.