Turbo tax made an error according to the IRS and now say I owe. I paid turbo tax to do the work correctly. Can I be reimbursed?

The IRS just generically refers to lots of things as "calculation errors"  and those things are often the result of errors made by the user who prepared the tax return.   If they changed your refund you will receive a detailed letter in several weeks that explains what was changed.  

Did you enter  ALL of the stimulus money that you had already received correctly in the recovery rebate credit section when you prepared your 2020 return?  The IRS cross checks and if you did not enter all of the money you received, they recalculate your refund.

MY REFUND WAS LESS THAN I EXPECTED

There can be a number of reasons why your refund was less than the amount you expected. First of all, look carefully at your actual tax returns.  There is sometimes confusion about the amount of refund to expect when users misinterpret the “summary” they saw in TurboTax. The amount of your refund is on your tax return.  Federal refund is on line 34 on Form 1040  State tax forms are all different so we cannot tell you what line to look at on your state form.  BUT it is important to actually LOOK at your tax return forms yourself and see the amounts of your tax refunds and/or tax due.  A lot of confusion would be avoided by doing that.

TurboTax does not receive feedback from the IRS regarding money that was offset from your refund.  If you owed back taxes, child support, student loan debt, etc. you will receive a letter of explanation from the IRS in several weeks.  The WMR site will contain a message regarding your offset.  It is also possible that the IRS caught a mistake you made and reduced your refund. 

A possible reason for the reduction could be that the recovery rebate credit was completed incorrectly and you may have entered the amount of stimulus already received wrong, which could have reduced the refund you thought you were seeing.

 It is also possible that your state reduced your refund for debts owed in your state—if the state reduced your state refund you will receive a letter from the state.

https://www.irs.gov/uac/tax-refund-offsets-pay-unpaid-debts

 IRS Treasury Offset Program Call Center at 1-800-304-3107

Be aware also that your Federal refund and state refund do NOT come together.  Federal and state tax returns are processed by two completely separate entities.  There is no rule as to which one will come first,

Did you choose to have TT fees deducted from your Federal refund?  (Remember that if you did that, you also agreed to an extra refund processing fee of $39.99)

How can I see my TurboTax  fees?   

https://ttlc.intuit.com/questions/2565973-how-do-i-review-my-fees-in-turbotax-online

 https://www.irs.gov/uac/tax-refund-offsets-pay-unpaid-debts

https://ttlc.intuit.com/community/refunds/help/what-is-a-refund-offset/00/26301

Reduced Refund      https://www.irs.gov/taxtopics/tc203

 NOTE: You can contact the IRS Treasury Offset Program Call Center at 1-800-304-3107 to ask if they have an offset for you on file. TurboTax would not have that information.

https://turbotax.intuit.com/tax-tips/tax-payments/who-can-garnish-an-income-tax-refund/L7cPPzDyc