This is pretty common for drilling through glass. Once you get to the end of the piece of glass, the glass tends to "break out" at the bottom instead of allowing the drill to cleanly exit the piece. It will happen virtually all the time and it's nothing you're doing that's causing it, it's just the nature of glass and stress.
You can, however, minimize or eliminate it from happening.
The easiest way to to take a scrap piece of glass and wax it to the bottom of your piece using stacking wax, or dop wax, or any convenient low melt wax you can find (parafin works well too).
Now when you drill through your glass, the core drill will continue on through the scrap glass and will "break out" on the exit side of scrap glass. Then you can take your piece and heat the area around the scrap glass to melt away the wax and separate your glass (you can even boil the piece if they are small enough to separate them).
This leaves a nice clean exit hole in your finished piece.