I remember about 1 month ago to do a bulk update on many tickets and there was an checkbox "Send mail for this update" (or something similar).
Today I created a bulk update to transition a bunch of items in the board but the option was not there. I see no way to avoid sending the notification email...
Have you removed that option or it depends on specific bulk operation and/or permissions? One month ago I tested with the same user and the option was there.
Here a screen capture of the confirmation step where the checkbox was placed
https://screencast.com/t/wdKZEkOb8