There's two things at issue here.
The first is zoom range, which is the longest focal length a zoom lens has divided by the shortest. That is, a lens which goes between 25mm equivalent focal length and 150mm is a "6x" zoom lens. This terminology is usually reserved for point and shoot cameras; for SLR lenses, one usually gives the actual focal lengths instead. High-zoom-range lenses require more design compromise, and it's likely that that compromise results in relatively weak performance at the extreme ends of the range. So that could be part of it.
Second is the issue of camera movement. Higher focal lengths — "more telephoto", or as you say, in the higher part of the zoom range — show a smaller portion of the scene magnified to the same size, and that means that small movements in the camera translate into larger movements in your photo. This means the effect of camera shake is much more pronounced the more you zoom in.
You can easily demonstrate this to yourself by simply looking at the live-view screen (or viewfinder) as you turn the camera slightly — at short focal lengths you can see a small change, and zoomed-in you can see that the whole scene changes with just a little turn. This same effect magnifies very small movements as well, increasing blur.
There's a particular compromise that most point and shoot cameras and superzoom lenses have which makes camera shake more of an issue when zoomed in. Specifically (as @Itai points out), these lenses usually provide a more-limited aperture at higher zoom. This means less light, which means either boosting the signal (higher ISO), resulting in more noise, or else longer shutter speeds — making it more important to reduce camera movement.
There's not much to be done about the first except to be aware of the strengths and weaknesses of your equipment, and to avoid using the higher focal lengths in situations where the weaknesses are most obvious — like in low-light.
For the second, simply keeping your camera more still will help significantly. You can get better results with improved technique and awareness of your motion as you press the shutter, but a tripod or other support will be even better. You'll also want to make sure that image stabilization is enabled in your camera if available — and make sure it has a chance to activate by half-pressing the shutter and waiting a second before firing.