The Quran does indeed pretty much mandate violence, as does the Bible, Tanakh, etc. The only real difference is the time and place in which said holy book is used to justify things. The Quran itself, as Enver Hoxha noted, is theologically superior to both the Bible and especially the Tanakh. Its contents are just about fully fleshed out both in the realm of the metaphysical and in the realm of government. For that reason it will be far more "violent" than either the Bible or Tanakh. It is simply a reflection of progressive versus reactionary in religious form, and the struggle between the two.
As far as debate over "violence" via mandated in the Quran goes, you generally get types like the Ahmadi, for example, who are materialists and believe that most Quran discussions dealt with situations of the time, and are not to be taken as always acceptable, but only acceptable under certain conditions. The Nation of Islam, Five Percenters, etc. follow this line of reasoning and it works well. "Extremist Islam" is simply using the Quran's logical conclusions, which are accepted by all Muslims who take their faith seriously, in a certain time or place in which violence is seen as necessary.
The logical conclusion of Islam is a society ruled by the Quran as interpreted through whatever governmental structure (in the Middle East and such, generally a Caliphate versus an Imamate). Just like the logical conclusion of Calvinism in some cases is that society is ruled by the elect who establish God's Kingdom on earth. To what extent one attempts to achieve this though, and of course the obvious interpretations that shift it towards liberalism, are highly varied. But Muslims are required to achieve an Islamic lifestyle, which does include having a government that serves Allah (or at least being protected from degeneration from a Pagan state). In the US, most Christians don't really take their faith seriously (and theologically Protestantism is lame), and religion takes on an essentially secondary (or just cultural) importance. Even the Christian fundamentalists have it take on a secondary nature, whereas types like Rushdoony and Van Til wanted it to be of primary importance in all walks of life.
As far as Jihad and such goes, it is simply a normal part of Muslim life. It calls for the defense of the faith against aggression and paganism, and to better oneself through the devotion to Allah. Interpretations vary, but it's generally accepted that preserving Islam and the piety of Muslims is the key component of that, which is what both "defensive" and "aggressive" (Qutbist) Jihad stress.
Hoxha wrote about the foundations of Islam (as he himself had been brought up in a devout Bektashi home) and on government in Islam: http://redrebelde.blogspot.com/2008/...ddle-ages.html
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