Filing Self-Employment (Private Contractor) shows that it will be "free" when adding deductions like mileage. Why am I charged $200?

The Self Employed version is not free.  I think it meant that if you upgrade to add your business expenses it might increase your refund more than the fee for upgrading.  

And do you know you somehow you upgraded to the LIVE version for higher cost?
How to remove Turbo Tax Live. There are 2 ways depending on how you got into LIVE.
https://ttlc.intuit.com/community/downgrading/help/how-do-i-remove-turbotax-live/00/27312

There is no Free online edition if you are reporting self-employment income and expenses.  However, if you are eligible the TurboTax Free File Program can be used.

The TurboTax online editions that can be used are the Self-Employed edition, $120, or the Live Self-Employed edition with CPA assistance, $200.

Your post indicates you are using the Live Self-Employed edition.

See this TurboTax support FAQ for checking your fees - https://ttlc.intuit.com/community/charges-and-fees/help/how-do-i-review-my-fees-in-turbotax-online/0...

See this TurboTax support FAQ for removing fees - https://ttlc.intuit.com/community/charges-and-fees/help/what-can-i-do-to-reduce-my-fees/01/26051

OR -

If you have not paid for the online edition you are using, have not filed your tax return or registered the Free edition, then you can clear your return and start over with a lower priced edition. Click on Tax Tools on the left side of the program screen while working on the 2019 online tax return. Click on Clear & Start Over.

OR -

Use the TurboTax Free File program if you are eligible - https://freefile.intuit.com/
If you are eligible you can use the TurboTax Free File Program which is free to file both a federal and state tax return. The Free File Program edition is a full featured personal tax program and on a separate website from the TurboTax online editions.

To qualify for free 2019 federal and state tax returns with the Free File Program, you'll need to meet at least one of these requirements:

• A maximum 2019 adjusted gross income (AGI) of $36,000; or
• You qualify for the Earned Income Tax Credit (EITC); or
• You served as active duty military (including Reservists and National Guard) with a maximum 2019 AGI of $69,000 (you'll also need a military-issued W-2).

See this TurboTax support FAQ on the Free File Program - https://ttlc.intuit.com/questions/1900583-what-is-the-turbotax-free-file-program

See this TurboTax support FAQ for the difference between the Free File Progam and the Free edition - https://ttlc.intuit.com/questions/2572617-what-s-the-difference-between-free-file-program-and-free-e...

And see this TurboTax support FAQ on how to switch to the TurboTax Free File Program - https://ttlc.intuit.com/questions/2026912-how-do-i-switch-to-turbotax-free-file-program