Practice turning pages very slowly with the choir. Make a game out of it. Without singing, turn pages and see how quiet everyone can be.
Find a way to stagger the page turns across (or even within) sections. You may be able to cut sheets of music in half and reassemble them in page protectors so page turns are staggered. (For example: the top half of page 2 now sits below the bottom half of page 1.) HALF of your binders would end up looking something like this:
|1|1|2|3|
|-|2|3|4| etc.
|1|1|2|3|
|-|2|3|4|
etc.
If you can't do that, you may be able to bring one system from the next page to the bottom of the one before. Even spreading all the page turns across a few bars will help.
You may find it quieter to put the music in page protectors and link all the pages together with plain binder rings--no binder. It looks a bit odd, but you're recording audio and not shooting a video.