I am exempt from paying self-employment taxes via the bilateral agreement with Japan. How can I claim this on my Schedule 2 through TurboTax?

You will not be able to do this in the Online version of TurboTax.  If you are using the online version, you can download your .tax file and finish your return in the CD/Download version.   See this article for help:  How do I save my TurboTax Online return as a tax data file?

To make the change in TurboTax CD/Download, please follow these steps:

1. Go into Forms Mode by clicking on the Forms icon in the top right of the blue bar. 
2. In the Forms in My Return list on the left, click 1040/1040SR Wks.  The form will appear on the right side of the screen.
3. Scroll to Schedule 2 - Additional Taxes.
4. In Part II right-click on the line 4 entry (Self-employment tax) and select Override from the menu that appears.
5. Delete the number that was entered.
6. You won't be able to type Exempt since it is a numerical field.  But when you print your return, you can enter the wording manually.
7. To return to the interview, click on the Step-By-Step icon in the top right of the blue bar.  

I purchased a desktop version as recommended, and followed the steps, and within the right-click context menu I do in fact see the Override option, but nothing happens when clicked. It appears disabled (i.e. it does not highlight when hovered over as Copy does). I can right-click and Copy, but I cannot right-click and Override. Please advise.

I think I figured out a workaround.

I clicked the magnifying glass icon that appears upon hovering to "goto supporting form" and was able to blank out the entry on the Schedule SE form (not the 1040 Schedule 2).