My supposition is that at the same price (in the 150-250$ price range) a lens with 10x or more zoom has worst image quality than a 4-5x. Is it correct?
Is usually quality of superzoom acceptable or not?
This is yet another question which borderlines on too generic. Given cameras which are otherwise identical and built with the same expertise, a longer zoom range has to be of lower quality because it simply has to do more but you will never find such cameras to compare. The cost is a combination of too many factors and the materials used in the camera are only a fraction of it.
Only you can judge if the quality is acceptable or not and it will depend on the situation. I own two ultra-zooms which I use for product photography which is never seen above 600 pixel wide. In this case, I prefer this than a DSLR because I get so much DOF thanks to the small sensor. I have even sold larger (20"x15") prints from a 7 year-old ultra-zoom, so clearly it can be considered acceptable.
Remember that getting the shot is often considered more acceptable than not getting it! Even when I carry a DSLR, I often have an ultra-zoom with me. With a 26X optical zoom lens, it can shoot just about anything quicker than I can change lenses on the DSLR for which only have short high-quality zooms and prime lenses.
Make sure you're comparing apples to apples. Some of those "low-zoom" cameras actually have some very nice sensors in them. A larger (physically) sensor can produce near-DSLR quality, but it makes it harder to support high zoom ranges because the optics start to scale up in size, too.
It's hard to give a general answer to this question, since I don't know which two cameras you might be looking at, but comparing something like Canon's S series (S90, S95, etc.) to their SX series (SX30, SX40 etc.), you'll see a larger sensor (1/1.7) in the S camera, vs. the superzoom (1/2.3). All things being equal (which, of course, they can't be without picking two specific cameras), you'd expect the larger sensor to produce a higher-quality image, pixel-for-pixel.
So, if it's possible to make a general statement about product lines, then yes - it's not uncommon to see superzoom cameras trade sensor size for zoom capability, and this can certainly affect image quality.