I paid $20,000 this year (2019) towards 2018 federal taxes. Where do I report that in Turbotax Home and Business?

Are you referring to estimated taxes paid in 2019 for tax year 2018?

Or are you referring to paying federal taxes owed for a prior tax year?

Estimated taxes paid in 2019 applied to tax year 2018

Were the estimated tax payments for 2018 paid on or before January 15, 2019?

No.   I paid $20,000 in March and April and allocated it to 2018 taxes.   

The IRS payment site permitted me to apply it to 2018 taxes.  Didn't say anything about being applied to the 2019 tax return.

The IRS tax payment website only allows estimated tax payments for tax year 2019 - <a rel="nofollow" target="_blank" href="https://directpay.irs.gov/directpay/payment?execution=e1s1">https://directpay.irs.gov/directpay/payment?execution=e1s1</a>

Or are  you referring to making payments for 2018 federal taxes owed on a 2018 tax return?  If so, those payments are not entered on the 2018 tax return.

So, if I made those payments this year to be applied to 2018 and then submit my tax return without reflecting those payments, does that mean I have to pay the amount again, knowing that I don't owe as much as the return shows?   Do I file for an extension, pay the actual net tax liability with the extension, and then file the actual return when the IRS shows that all taxes due have been paid?

The IRS already knows you have made payments for the 2018 taxes owed.  If you still have a balance after making those payments then you can pay the balance with the 2018 tax return that is being filed.  And if filed and paid on or before April 15, 2019 there will not be a failure to pay penalty.
You are Not "required" to pay what is shown on your tax return for the taxes owed when filing the tax return.

Estimated tax payments for tax year 2018 had to be paid on or before January 15, 2019.  Taxes paid after that date can only be applied to the 2019 tax return.

To enter, edit or delete estimated taxes paid (Federal, State, Local) - 

• Click on Federal Taxes (Personal using Home and Business)
• Click on Deductions and Credits
• Click on I'll choose what I work on (if shown)
• Scroll down to Estimates and Other Taxes Paid
• On Estimates, click on the start or update button

Or enter estimates paid in the Search/Find box located in the upper right of the online program screen.  Click on Jump to estimates paid

Or click on Tax Tools on the left side of the screen.

• Click on Tools
• Click on Topic Search
• Type in estimated tax payments, click on GO

Federal estimated taxes paid are entered on Form 1040 Schedule 5 Line 66.  The totals for Schedule 5 flow to Form 1040 Line 17 Federal credits