You should not have to start your return from scratch to enter your self-employment income.  Try using the approach described below to get your Schedule C and Form 1099-NEC entered into your tax return.  

TurboTax will guide you through the process by asking you questions one step at a time.  

Use these steps to go to the Schedule C section of your return:

• On the top row of the TurboTax online screen, click on Search (or for CD/downloaded TurboTax locate the search box in the upper right corner)
• This opens a box where you can type in “schedule c” (be sure to enter exactly as shown here) and click the magnifying glass (or for CD/downloaded TurboTax, click Find)
• The search results will give you an option to “Jump to schedule c”
• Click on the blue “Jump to schedule c” link

Once your Schedule C is set up, use the Add Income option to enter the details from your Form 1099-NEC.

Take a look at the following TurboTax article for more information:  Does a 1099-NEC or 1099-MISC mean I'm self-employed?

This solution leads me back to the same issue.

Whenever I click "Jump to Schedule C," it takes me to the same page that says "Did you have any self-employment income?"   When I click "Yes," the page still only reloads, and gives me the exact same question without ever giving me an option to enter my 1099-NEC.   No matter how many times I click "Yes," it just links back to the question "Did you have any self-employment income?"

For context, over the past few weeks, I have tried logging out and back into my account numerous times, logging in at different hours and days to make sure it isn't a traffic issue on the website, reset all the cookies in-use, and disabled any ad/pop-up blockers for the site.   Despite all of this, I still have yet to have the "Yes" button on the "Did you have any self-employment income?" take me to any other page.

What version of TurboTax are you using? Usually you should see an upgrade screen if you are in a lower version that does not handle self-employment.

Rather than using the Jump to feature try clicking starting Self-Employment Income and expenses in Wages & Income / Self-Employment.

Or go to Other Common Income and select Form 1099-NEC.

I am using the TurboTax Free Edition through a browser on my laptop.   It looks like this must be my issue, though I should also note that clicking the "Upgrade" button doesn't do anything for me either...

I am going to find a different way to file those specific taxes separately – the only reason I have a 1099-NEC in the first place is because I briefly took up a delivery service job at the onset of the COVID-19 pandemic when my primary career took a massive hit (and has since largely recovered), and I don't plan on picking that supplemental job back up again in the future, so paying over $100 for a one-time use feature doesn't make logistical sense to me.