Federal and state refunds come from completely separate entities, There is no rule as to which one will come first or how much time there will be between their arrivals.
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.
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 20 of your Form 1040) of your federal refund to track your Federal refund:
Did you have health care credits? TT did not file the necessary health care credits form for me - and I have been informed by IRS that my return was rejected and that I will be ineligible for health care credits in the future for not filing that form.
If this happened to you, do not waste your time trying to get help from TT - they will only tell you it is your fault; that their money back guarantee does not apply to you and that (in my case) even though my income was well below 34k, I still had to pay $75 to have my return filed.
Do complain to the IRS and your federal representatives. TT knows they have messed up royally but they will not own up to it ... yet.