Although you expect your bathroom to hold some moisture, that dampness in the air can create problems for your light fixtures. The fixture sockets and the threads on light bulbs are metal, and prolonged exposure to dampness can corrode them, causing the bulbs to get stuck in the sockets. You have to exercise some care when removing a stuck bulb so you don't cut your fingers and litter the floor with glass. Once it's out, a little lubrication should make it easier to remove in the future.
Turn off the switch that controls the light fixture.
Unroll the end of a roll of duct tape, fold it back on itself and stick it to the roll so that it forms a sticky pad.
Place the pad on the end of the stuck bulb and turn the roll counterclockwise. The bulb should turn with it. If it won't turn, spray lubricant between the threads of the bulb and the inside of the socket. Wait for a few minutes, then try turning the bulb again.
Spray lubricant around the inside of the socket and on the threads of the bulb after you have removed it. Screw the bulb back in, then unscrew it again to spread the lubricant over the threads and dissolve rust.
Remove larger rust deposits with a rag soaked with white vinegar. Wipe the inside of the socket with the rag thoroughly, then wait for about an hour and rub it again. Continue in this way until the rust is removed.
Clean the socket with a solution of 1/2 tsp baking soda to 1 cup of water to neutralise the vinegar. Let the baking soda solution dry thoroughly before spraying lubricant on the socket and screwing the bulb in.
Use a potato to remove the bulb if it breaks. Cut the potato in half and push one half against the remaining glass or the metal threads. Turn the potato counterclockwise and the bulb will turn with it.
If the corrosion is advanced, it may interfere with the electrical connection, and the bulb may flicker. If this happens, remove as much corrosion as you can with vinegar or a wire brush, and use a new bulb.
If corrosion is causing your bulb to flicker, repair the fixture as soon as possible. The poor connection may be causing high-temperature, hazardous arcing inside the socket.