Accepted means that you passed the basic identity verification with the IRS.  Once your return gets accepted then only the IRS knows when they will issue your refund.  TurboTax does not get any more information after your return has been accepted.  Since the IRS did not open their filing system until 2/12/21, they are busy & it can take longer for your return to be processed.  If your return includes EIC, then the IRS states that in the beginning of March they should be issuing refunds.

Once your returns have been accepted, the IRS states that nine out of 10 e-filed tax returns with direct deposit will be processed within 21 days of IRS e-file acceptance. Mailed paper returns: If you filed a paper return, please allow 4 weeks before checking the status. Refund processing time is 6 to 8 weeks from the date the IRS receives your tax return.

Also:  The EIC/Additional EIC was not processed with the IRS until later in Feb., so it would be longer than the 21 days since you filed early.

From the IRS:

If you claimed the Earned Income Tax Credit (EITC) or the Additional Child Tax Credit (ACTC), you can expect to get your refund by the first week of March if:

• You file your return online
• You choose to get your refund by direct deposit
• We found no issues with your return

Additionally, your financial institution may need time to accept your direct deposit or issue a debit card. Many institutions don’t process payments on weekends or holidays. So, if you file early, be aware of federal and local holidays that may affect how soon you get your refund.

You can checked the IRS website Where's My Refund for your information?

https://www.irs.gov/refunds

Please note that the IRS website, Where's My Refund, has not been updating as it should, so you might have your refund in your bank account even though there is no update to the IRS website.  It happened to my son, his deposit was in his bank account even though the IRS website still showed received.. So you might want to check your bank account.

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How do I track my New York tax refund?

https://ttlc.intuit.com/community/refund-status/help/how-do-i-track-my-new-york-tax-refund/00/26586