I paid my Illinois state taxes using my credit card via iltaxpayment(dot)com. My payment was received and processed immediately. When will this reflect on Turbotax?

The payment will not be reflected in TurboTax. TurboTax shows the amount you owed as of when you filed your tax return. TurboTax does not get any information from either the IRS or the state about payments that you make.

It will never be shown in your TurboTax account.  Your online account only shows the result of the return you prepared using the software.  TT never knows if, when, or how you paid your tax due to the IRS or the state.  Your own bank/credit card records are your proof that you paid.  Store the records somewhere safe where you will not lose them.

if your payment was not an extension payment ignore the rest.   TT not does know and does not include in the return extension payments made outside of the program.  so if this was an extension payment you need to enter this payment in the tax payment worksheet.