While your bathroom does not need to be the brightest room in the house, there is still a need for it to have decent lighting especially around the bathroom sink. Of course it does not need to be as brightly lit as the kitchen for example, or your living areas where you read, though. There are simple ways to improve your lighting that do not cost a lot of money. You can either put new, modern lighting in your bathroom or maybe rather than installing a whole new lighting system you could simply just change your current bathroom light fixtures.
Before you go to all the trouble of doing that, though, why not just go ahead and wipe off your light fixtures. It is amazing how often simply taking a damp cloth and cleaning your fixtures can improve the light situation in your bathroom. Dust really does hinder the amount of light that comes through a fixture.
Reassess your bathroom sink area first. Most likely this is the area where you feel you need the brightest light as it is used for a lot of detail tasks such as shaving or putting on makeup. Overhead lighting tends to make shadows on the mirror, so your best choice is to add bathroom wall light fixtures either at the top of the mirror or around the sides of the mirror. Natural type lighting is the best to use for this area.
For the rest of the bathroom you can use softer lighting, and it does not need to be natural light either. Yellow or softer bulbs and lighting work great for the main lighting in the bathroom. If you do not want your entire bathroom lit up like a hospital foyer you can always install a dim switch or just put in recessed lighting, or have a single main light and then recessed lights in the shower area. Incidentally, it is important to make sure that the shower area is decently lit too, just not with harsh lighting.
If you do not want to go to all the trouble of installing new lighting, maybe the combination of cleaning your light shades and making sure light can get through them, and then taking the curtains off your windows and replacing them with opaque paper at least on the bottom part of the window, may be all that is needed.