Why is IRS saying that taxes haven't been paid, when they were paid through Intuit Turbotax and charged on the credit card? Did Intuit Turbotax not pay the IRS?

No.  TurboTax does not pay your tax.   TT can only pass your banking or credit card information along to the IRS and the IRS has to pull out the payment.   TT does not touch your payment at all.   Do you have a bank or credit card statement showing that the payment was made?   What does it say on the page of filing instructions that prints out with your tax return?  What does it say about how you are paying your tax due?   

If the IRS is saying that you did not pay, and you have proof that you paid, then you sort it out with the IRS.