Welcome to iPhoneForums!
The only way to mute the shuttersound on an iPhone is to use the Mute button next to the Volume buttons.
This would also mute system sounds, afaik, so you wouldn't hear texts and emails as well.
It seems that the behaviour on your iPhone is unusual.
Try resetting your iPhone: press and hold Home and Power button until the Apple logo appears, then let go of both, the iPhone will power up on it's own.