I’ve recently paid the IRS the amount I owed, but it still shows that I have to pay. How can I see if I paid it correctly ?

The program will ALWAYS show your return as filed with the balance due on the bottom line ... TT never gets told if the bill was actually paid.  Your only proof of payment is the cancelled check or debit on your bank or credit card statement.  

The TurboTax Tax Home website will show that the taxes are owed.  TurboTax has no way to know if or when the taxes owed were ever paid.

Your cancelled check, bank or credit card account statement with the amount paid and showing the Payee as the IRS or US Treasury is your proof of payment.