Discomfort is inevitable when you start out as Hunter60 says, but it should pass. I'd be willing to bet you're holding on to the neck to tight to try to get all the notes out and stop the buzz. Hold a barre chord and look at your fingers/thumb - your nails probably look white because you're pressing so hard.

Try backing off a bit to find out exactly how much pressure you need to use. The strength in your hand to apply barre chords should come from your fingers rather than pressure between fingers and thumb.