How will they know that one of my distributions from a 403B was a rollover?

Rollovers get reported on Form 1099 R, with a code G in Box 7. You should get a Form 1099-R from the custodian which reports the rollover. You will have to report your Rollover on your taxes.

To enter the information in TurboTax Online, follow these steps:

1. Type “1099-R” in the Search box.
2. Select the "Jump to" link in the search results.
3. Enter the information from your 1099-R form and continue.
4. At the What did you do with the money? screen, select I moved the money to another retirement account.
5. From the drop-down list, select the rolled over option and continue.

If taxes were withheld in Box 4 of your form, they'll either be added to your refund or applied to your tax liability.