Re- read all of that, and LOOK at your federal tax return.  The amount you owe to the IRS is on line 23 of your Form 1040.  

You got confused by the summary that TurboTax shows you.  The difference between the amount you owe the IRS and the amount you got back from the state is $493.  

https://ttlc.intuit.com/community/prior-year-return/help/how-do-i-get-a-copy-of-a-return-i-filed-thi...

If you owe federal tax it is due by July 15, 2020.

If you have federal tax due you can pay by mailing your payment with the 1040V voucher, having the payment taken out of a designated bank account, or you can pay directly on the IRS website.

https://www.irs.gov/payments/direct-pay

https://ttlc.intuit.com/community/tax-payments/help/how-can-i-pay-my-federal-taxes/00/26212

https://ttlc.intuit.com/community/tax-payments/help/how-do-i-pay-my-irs-tax-due-with-a-check-or-mone...

The $5044 is the amount of federal due of $5537 less the state refund of $493. You owe $5537 for your federal taxes. That is the amount you must pay. 

ok. thank you.    I see line 23 says clearly 5537.   But so why is Turbo cover page showing  5044 due.  it's confusing and I don't want to underpay the gov't.    (I'd actually prefer to pay the lesser amount)!

Oh, I see now. Thanks for clarifying. That was helpful.    oh well.