BlueTurn is not an Airturn device. (We are not aware of any third party products using the Airturn protocol but they might be out there.)

The iPad and iPhone versions of iReal Pro respond to left/right arrow for page turning.
[Some Airturn modes provide left/right arrow also, so it might (should?) be possible to still have Airturn on and have BlueTurn function, but obviously you both have BlueTurn working with Airturn OFF, which is what we would have expected.]

If you have an Apple keyboard, there are other key commands other than left/right arrow you can use - change tempo, next/previous style, tap tempo etc. as well as making it easy in the Editor for inputting and editing songs.

Thank you for your replies, we now know BlueTurn works again (sorry for the bug, it was a fault with the left/right arrows)
Fixed in iReal Pro v2022.2 (iOS)