Can someone explain why the amount I owed in state taxes was deducted from my federal refund and I still had to pay the state tax?

State taxes owed can NEVER be paid from a federal tax refund.  Only the TurboTax account fees can be paid from the federal tax refund if you chose that option to pay the TurboTax account fees.

If you did not pay the state taxes owed when you filed your state tax return, then those taxes are still owed to the state.

Look at the State Filing Instructions page for how you chose to pay the state taxes owed.

To access your current or prior year online tax returns sign onto the TurboTax website with the User ID you used to create the account - https://myturbotax.intuit.com/

When you sign onto your online account and land on the Tax Home web page, scroll down and click on Add a state. 

This will take you back to the 2021 online tax return.

Click on Tax Tools on the left side of the online program screen.  Then click on Print Center.  Then click on Print, save or preview this year's return.  Choose the option Include government and TurboTax worksheets

If your federal tax refund has been reduced by the IRS, you will be receiving a notice from the IRS explaining the changes to your tax return.

Does the IRS website indicate that you were sent less than what was reported on your federal tax return Form 1040 Line 34? - https://www.irs.gov/refunds

Did you get the federal refund?  Was it missing the 600?  At the end Turbo Tax just shows you the net federal and state combined. But that is only for your info. You will still get the full refund and need to pay a tax due. 

Your only proof is your bank account statement or credit card statement showing the payment coming out.  And Turbo Tax doesn't pay any tax due for you.  Turbo Tax just sends your payment method to the state.   You either put down your bank account number for the state to pull it directly out of your account or you needed to mail in a check.  Most states don't let you pay by direct debit.  Did you check your bank account to see if it came out?  You should have an instruction sheet telling you how to pay it.

Oh, and you can't have a state tax due deducted from your federal refund.  Never.  You can only have the Turbo Tax fees deducted from your refund.  See,  Can I pay my state taxes from my federal refund?

Check the printout or PDF of your return; look for the state cover sheet with the Turbotax logo.  If you owed tax, it will show the payment information and how/when you decided to pay.  Read the state payment instructions carefully since most states cannot be paid from within the TT system and requires additional steps.

https://ttlc.intuit.com/community/refunds/help/what-are-my-options-for-getting-my-state-refund-or-for-paying-any-state-taxes-i-owe/00/25954