TurboTax does not receive any kind of tax refund, federal or state.
If you have a tax refund, the refund can either be direct deposited into a US bank account in a bank located in the US. If you did not request direct deposit then the refund will be a check mailed to the address on your tax return.
To check the status of an e-filed return, open up your desktop product or log into your TurboTax Online Account. You can find your status within the TurboTax product.
If accepted by the IRS use the federal tax refund website to check the refund status - https://www.irs.gov/refunds
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
After the tax return has been Accepted by the IRS (meaning only that they received the return) it will be in the Processing mode until the tax refund has been Approved and then an Issue Date will be available on the IRS website.
See this IRS website for federal tax refund FAQ's - https://www.irs.gov/refunds/tax-season-refund-frequently-asked-questions