When you have a hammer, every problem looks like a nail. And so, liberals believe that the solution to every problem is more socialism. Americans often believe the solution to every problem is more democracy. And Muslims believe that the solution to every problem is Islam. Combine the three and you arrive at the inevitability of Sharia law in the West. As Muslims harness democratic pluralities in countries that have become socialist and thus less free, they will impose Islamic law.
In the conception of religion held by Western Liberals, religious moderates are people who are willing to allow the separation of religion from civic life or even its domination by the civic code. Religious extremists they believe are people who want to impose religion on public life. By this standard however, there is virtually no such thing as a Muslim moderate, because Muslims do not recognize the hallowed ideas of Western liberals such as pluralism and the separation of church and state.
Even the few exceptions such as Turkey, did not create separate spheres, so much as they imposed forced secularism in order to modernize the country. And these exceptions are also collapsing, notably in Turkey where the Islamists under Erdogan have come to power. The imposed secularism in countries such as Turkey originated at a time when it was thought that a Muslim country had to forcibly secularize in order to enjoy the benefits of a modern state.
But the willingness of Westerners to accommodate Islam and the billions in oil money that have flowed into Saudi Arabia and the UAE have discredited that notion by showing that one can be a fanatical Muslim and still be a doctor in England, or own skyscrapers in Dubai, be a Lord or a Peer, a respected professor in a French university or have a nuclear reactor assembled in your country. Political correctness, appeasement and dhimmitude have eroded the gains made by secularization and helped radicalize Islam.