Are window air conditioner support brackets designed to support a window air conditioner by itself?
The bracket is entirely sufficient to support your AC.  That's doesn't mean that in some freak occurrence it couldn't come out of the window. If you want to add to the margin of safety, you want to prevent the upper sash from loosening enough to prevent it from rotating out. Most units have screw holes in the frame of the AC that allow you to fasten it to the upper sash.  Otherwise a small screw or nail to prevent the sash from moving will do the trick. At that point not much short of a tornado is going to get it out of the window...

Hope this helps!