Turbo take appears to have misled me into thinking my IN-state tax return was successfully filed and accepted. I just received a demand notice for payment from IN DOR

If it was not accepted then the state wouldn't know you owed.

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

So if I understand your response correctly, you are saying that the Department of Revenue in Indiana had to have received my electronic payment via turbotax since they new to send me a notice that they didn't receive it.  But why are they sending me the notice if they received my payment?

Correct ... the state return was successfully efiled if they are billing you for the unpaid balance due. 

So do you have proof of payment to the state  in your bank account ?   If so then send them proof of payment in response to their bill.  

If you do not see proof of payment then you need to send it now ... as already mentioned review the state filing instruction page to see how/when you were to pay the balance due. 

I checked my bank account and I do not see proof of payment.  I assumed Turbotax paid since I received confirmation from Turbo tax that it was successfully filed with the state.

Are you sure you included your bank information (routing number, account number, etc.) on your state tax return?

@crstuart - just because you efiled, doesn't mean you gave Turbo Tax authorization to debit your checking account.   

These are separate authorizations in Turbo Tax: one permitting efiling and one permitting debit of your checking account, 

You must pay your state tax due using the state’s preferred method of receiving payment.  For most states that will be by making a payment to the state’s own tax website, or by mailing a check or money order.

https://ttlc.intuit.com/questions/1900884-what-are-my-options-for-getting-my-state-refund-or-for-pay...

https://ttlc.intuit.com/questions/1901670-how-do-i-contact-my-state-department-of-revenue

How do I review the method I chose to pay my federal income tax with, after filing with TurboTax Online?

You can verify the tax payment method you chose on the Electronic Filing Instructions for your 2021 Federal Tax Return page, next to Balance Due/Refund. To see the page, follow these instructions:

1. Open your return if it’s not already open.
2. On the [Your name], your 2021 federal return was accepted! screen, scroll down and select Add a State (don’t worry, we’re not adding a state) to access your tax forms and worksheets.
3. On the left side select Tax Tools then Print Center.
4. On the TurboTax Print Center screen, select Print, save or preview this year’s return.
5. On the next TurboTax Print Center screen, select 2021 federal returns and Include government and TurboTax worksheets (optional) then select View or print forms.
6. The information will be on the first page, Electronic Filing Instructions for your 2021 Federal or State Tax Return next to Balance Due/Refund.

@Critter-3   is correct. Some states allow the debit payment to be enabled when filing the tax return, while many others don't.   It would be nice if we had an FAQ list of which states allow it , and which don't, but it can change every year.  NC just started allowing it a couple years ago.

In the past years, Indiana has required the payments for balance due, to be submitted entirely separately and must be done by the taxpayer separately from their e-file..  Apparently that sounds like it is true for Indiana's 2021 taxes  too, but they (Indiana) could change that any future year.

Sadly some users fail to read the screen in that section which tells them exactly how they are choosing to pay the state balance due  and/or  the filing instruction page before they file.  Personally I think that downloading the PDF of the return should be mandatory along with a check box that says something like "I have thoroughly reviewed the return & I believe it to be correct and accurate"  and if there is a balance due " I have thoroughly reviewed and understand the way I have chosen to pay the balance due"  to at least make people aware of how important it is to check everything then check it again. 

I also believe there should be a mandatory "holding period" before someone can efile "say overnight or at least a few hours)  as we see too many folks use the phrase "I just filed and found a mistake how can I correct it ?"