The IRS does not directly accept credit cards for tax bills. The credit card number is passed to a third party processor, who charges your tax plus a fee, and then sends the tax money to the IRS. It is possible that the processor used by Turbotax does not accept Amex for income tax payments, even though Intuit accepts Amex for payment of program fees.
Instead of paying through the program, select "I will pay by mail" and file without a payment. Then, go to https://www.irs.gov/payments and select any convenient method, you can choose bank debit or a variety of credit card processors, some of whom may accept Amex. Just be sure to make the payment before April 15.
