Federal and state refunds come from completely separate entities.  There is no rule as to which one will come in first or how long it will be between their arrival in your account.

Sadly, the IRS refund site has been behaving erratically and has not always been reliable this year.   TurboTax has no control over this and cannot find out anything about your refund for you.

https://www.irs.gov/refunds

https://www.msn.com/en-us/money/taxes/the-irs-is-behind-in-processing-nearly-7-million-tax-returns-s...

TurboTax gives you an estimated date for receiving your refund based on a 21 day average from your date of acceptance, but it can take longer.  “21 days”  is not a promise from TurboTax or the IRS.   Many refunds are taking longer during the pandemic.  The IRS is backlogged —-they are still trying to process millions of 2019 returns; they received millions of e-filed on the day they opened for 2020 returns, and now they are burdened with sending out another round of stimulus checks.   

Some delays are resulting from incorrect amounts that folks entered for stimulus checks they received.   Many people have been confused by the recovery rebate credit and completed it incorrectly.  The IRS may delay your refund while they cross-check to see what you received.

First, check your e-file status to see if your return was accepted: 

https://turbotax.intuit.com/tax-tools/efile-status-lookup/

Once your federal return has been accepted by the IRS, only the IRS has any control.  TurboTax does not receive any updates from the IRS. Your ONLY source of information about your refund now is the IRS.

You need your filing status, your Social Security number and the exact amount  (line 34of your 2020 Form 1040) of your federal refund to track your Federal refund:   

https://www.irs.gov/refunds

To track your state refund:    

https://ttlc.intuit.com/questions/1899433-how-do-i-track-my-state-refund

https://ttlc.intuit.com/questions/1901548-why-do-some-refunds-take-longer-than-others

If you chose to have your TurboTax fees deducted from your federal refund, that will take some extra time, while the third party bank handles the refund processing.

https://www.irs.gov/refunds/tax-season-refund-frequently-asked-questions

https://ttlc.intuit.com/questions/2840013-does-accepted-mean-my-refund-is-approved

https://ttlc.intuit.com/questions/2724106-my-return-is-accepted-but-still-not-approved-is-there-a-pr...

No, there isn't anything that you need to do. The IRS is currently taking longer to process returns. Please see What if my IRS refund is taking longer than 21 days? for information what to do and why it might take longer.

Also, you can track your refund at the IRS Where's My Refund? site.