Is it normal for an Inner tube to bulge when inflated? I recently bought a pair of inner tubes and when I partially inflated one of them before putting it into the tire (only a few pumps with a floor pump), I noticed a portion of it was bulging out considerably. It looked like a snake who had swallowed a giant mouse, there was a huge bulge on one section of the tire. I went to get my camera to take a photo of it, but it then burst before I could take a photo. 

Went back to the LBS and got a replacement tube, and when I partially inflate this one it also bulges quite a bit too, although not nearly as extreme as the last one that popped. 

Is it normal for an inner tube to bulge out as much as this? I'm just worried this one will pop as well.
Yes, inner tubes do that. They are like balloons, except that the butyl rubber they are made of doesn't stretch like natural latex rubber. 

You are not supposed to inflate tubes outside a tire. Tires have fabric casing to withstand the pressure.