How do I reflect that a distribution from a 403b retirement account was reinvested into another IRA account within 60 days?

This is a rollover. You indicate that you have done a rollover when entering your form 1099-R.

1. In TurboTax, open your tax return
2. Click on Federal in the left-hand column, then on Wages & Income on top of the screen
3. Scroll down and locate the section Retirement Plans and Social Security and click on Show more
4. Click Start next to IRAs, 401(k), Pension Plan Withdrawals (1099-R).
5. Enter your form 1099-R and follow the interview
6. When you arrive to the page Tell us if you moved the money through a rollover or conversion, click on the radio button I rolled over some or all of it to an IRA or other retirement account within the time limits (normally 60 days)