My computer's in the shop at the moment and I'm working on a PowerPC with an outdated version of iTunes, but I'm going to try to work from memory.
The one way I know will put an end to this is a setting I believe is in the preferences menu of iTunes. I'm going to guess it's under Devices (has an iPhone icon) and should say something like "don't open iTunes or other software when connected."
Personally I'm not a fan of Adobe Bridge and I don't have a lot of experience with it.
Of course halfway through writing this... do you mean moving the photos from your computer to your phone or vice versa? If it's moving them from your phone to Bridge, I'd imagine there should be a setting to not import duplicates. Again, I don't use Bridge so I don't have a good idea of the layout, but if it's meant to manage importing photos, it has to have a feature like that.