What Does it Mean to "Islamize the West?"
"...eliminating and destroying the Western civilization from within and "sabotaging" its miserable house..." Read more about that here.
The Muslim Brotherhood has established what appear to be legitimate, mainstream, moderate Muslim organizations (opulently funded with Saudi oil money) which have set themselves up as representatives of "the Muslim community" to approve or disapprove of what is in American textbooks, what is published in magazines, newspapers, and television news, and how Islam is portrayed in Hollywood movies. These organizations have been able to influence how the U.S. governmental security agencies think about and write about Islamic terrorism. They are successfully silencing criticism of Islam in the West. And in numerous other ways, orthodox Muslims are getting Islamic law applied in the West, and getting it applied more completely with each passing year.
Because the percentage of Muslims is much higher in Europe than in the United States, Europe is further along in the process of becoming "Islamized." It's hard to believe, but European laws are giving way to Islamic law. Muslim pressure continues to find cracks, and it works its way in. For example, all over Europe, the right to free speech is giving way to Sharia's limits on free speech (source 1 and source 2). Belgium, Germany and Britain offer benefits to polygamists' wives, even though polygamy is illegal there (source). Schools in Amsterdam don't mention farms because pigs are offensive to Muslims. "Many state schools in Belgium and Denmark only serve halal food to all pupils. In France, school teachers are advised to avoid authors deemed offensive to Muslims, including Voltaire. The history of the Holocaust can in many cases no longer be taught because of Muslim sensitivity (source)."
France is allowing Muslims to rule parts of their country (source). Britain allows unfair treatment of Muslim women (source). Muslim doctors in Britain don't have to follow the same rules of cleanliness as everyone else (source). Britain knowingly tolerates sedition by Muslims (source). Britain allows over 80 Sharia courts to operate within their borders (source). In these and many, many other ways (source), the laws, values, and principles of Western civilization are giving way slowly but surely to unceasing Islamic pressure.