You can track it at the New York State Department of Taxation and Finance Check Refund Status page. You don't need an Online Services account to use the Check Refund Status page.
The state will ask you to enter the exact amount of your expected refund in whole dollars. You can find this by logging into TurboTax. If you're using the mobile app, you may need to scroll down to find your refund amount.
The refund date you'll see doesn't include the days your financial institution may take to process a direct deposit or a paper check may take in the mail. If you're concerned, contact the New York State Department of Taxation and Finance.
As much as we would like to help, TurboTax does not distribute refunds, only the IRS or your state tax agency can legally do so.