Turbo tax shows my return not only accepted but a deposit date as of this past march 3rd but, the irs website is not even showing accepted….. why?

Any date provided by TurboTax is an estimate based on the common 21-day average.   It's not uncommon for return processing to take longer than the estimated 21 days.   You should only go by any date if it's provided on the IRS website. 

"Accepted" is an efile term that means the efile was in the proper format such that the IRS agreed to begin processing it.   It is not a term used on the IRS website.  The IRS website should initially say something like "received" and/or "processing."

See what is currently displayed at the IRS "Where's My Refund" tool.  Sometimes the IRS will leave a message if more info is needed or if they are preparing a letter, etc.
https://www.irs.gov/Refunds

• Be sure you have waited at least 24 hours after acceptance before using the IRS WMR tool.
• Be sure you enter the correct SSN and filing status.
• Be sure you are using only the Federal refund amount.   Do not include any state refund or any total refund or net refund (Fed and State) that TurboTax may have provided on a screen.  
• Also be sure you are using the Federal amount before any TurboTax fees are taken out (if you chose that option.)
• Look at your actual Federal return to get the Federal refund amount, i.e., your Form 1040, Line 35a.

FAQ:  How do I track my state refund?

https://ttlc.intuit.com/community/refund-status/help/how-do-i-track-my-state-refund/00/25571