Here’s an old electrician’s trick that might work for you. Don’t laugh when you start to read it, I’m serious. Take a firm raw potato, peel it and carve it down to a cylinder small enough to fit up inside where the glass is broken off the base of the bulb. Push the potato in firmly against the base and then try turning it. If you’ve got a chunk of styrofoam this can work, too. The idea is to have a piece of soft material that the rough edges left around the bulb filament can catch on so you can get leverage to turn it. I hope you are making sure the switch is off before you mess around with this.

The vaseline suggestion above is a good one for a bulb in any area of high moisture. I would question the wisdom of using a halogen lamp of any kind in a bathroom though — these operate at very high temps and can cause burn injuries if they break. Some can even explode. Also, the whiter light emitted by halogen is not very flattering to your skin looking in the mirror. Whatever you decide to install for a new bulb in this fixture, buy a good brand name one like GE or Phillips. The cheaper Chinese made bulbs will tend to fall apart like the one you are having the problem with.