Yeah, it is normal. Because there is some distance between your eyes and the glasses while the contact lenses are worn near your eyeballs, the prescription between the two types of glasses must be different. According to the physical refraction principle, the prescription of contact lenses must be lower than that of frame glasses. Thus, the effect may be the same for the person with shortsighted eyes.
Yes, it is common to have a lower contact prescription than your glasses prescription. The contact prescription is lower because the contact sits right against your eyeballs, but the glasses sit about 1 inch away from your eyeballs. In other words, when decide the prescription of the contacts or the glasses, the first thing you should consider is about the distance between the lenses and the eyeball. But you can not decide the prescription by yourself but through a serious exam and the doctor's suggestion.