What is wrong with democracy?
That depends on who you are.
Why would extremists like democracy? By the very fact that they are...extreme...their opinions will not be approved of by the populace at large. Extremists cannot use a democratic political system to enact their views into law.
Other than that, some of those who take to terrorism do so because they perceive (rightly or wrongly) that the political situation is corrupt and/or cannot be influenced.
Look at the two most notable terrorist acts here in the U.S. in the past few years.
Oklahoma City: The perps in this incident had no reason to believe their views would be supported by our democratic system, because they were suspicious of gov't to the point of paranoia. From their point of view, petitioning the gov't for redress was like asking a fox to guard the henhouse.
911: The perps in this incident were upset by Western (and particularly American) colonial interference in their native lands. They are not happy about their own gov'ts either, but there are a number of reasons why they might choose to hit us first. Oh sure, everything is couched in terms of Islam, but don't make the mistake of thinking they would be attacking us if we didn't have a foreign policy designed to keep their native land oppressive and corrupt and ensure we maximize the profits of certain people living here. bin Laden has been very up front about what he didn't like -- U.S. presence on Saudi soil. Look, if a foreign country had bases here, would *we* be happy about that? I doubt it. He's an extremist to be sure, but he's not completely lacking in reason. To believe "they hate our freedoms" is to make the cardinal mistake of believing our politicians are actually telling us the truth all the time.
The world is so varied and getting smaller everyday and not all of us desire to be controlled in a manner that Sharia law would impose.
There are many versions of what is sharia law, and they differ greatly from one culture to the next. It would be a mistake to assume that some of the things you hear about what's in sharia law applies everywhere. (read the Islam forum here for some ideas about this)
Think of the situation in Judaism, for a similar example. What "the Law" entails depends very much on whom you consult. A Reform Jew might tell you that he will never, ever eat a kid (goat) boiled in its mother's milk. An Orthodox Jew would tell you he will never, ever eat a cheeseburger. They don't have a lot of strife over differences, but still there are a few neighborhoods in the world I would not drive through on Shabbat. My car might be pelleted by rocks, even though I am not Jewish and am not subject to the same Law they are.
Religions are not different than other organizations -- they all have their extremists.
Even mine does. We're just fortunate enough to have an organizational system that dampens and dilutes and does not encourage extremist views. Oh, that might change some day, but in the meantime I'm enjoying the peace and quiet. :beach: