How do you get an Ottoman Empire that is fairly liberal, staving off the more radical/conservative aspects of Islam? After all it's ruling exactly the regions of the world that turn pretty radical IOTL but I think the Ottoman Empire gets underestimated on the board and would like to hear ideas of how it handles the issue.
(Put aside the issue of it surviving, and let's think fairly recent—19th century or so, but if this better belongs in the the After 1900 forum it can always get moved.)
If the empire survived, it would have been. Radical Islam was not exactly a problem, nor was it particularly conservative. There were of course conservatives, like there are everywhere, but the prevalent image of some rabidly paleolithic clerical class trying to keep the empire Medieval is a relic of Victorian biases. The clergy was instrumental to the modernization process, and many of its members were the foremost reformers.
There were also religious conservatives, like there are everywhere, but in general the late Ottoman Empire was more progressive than most of the Middle East today.
Part of the reason for that is that the British and French consciously retribalized the region, undoing centuries of Ottoman work to sedentarize nomads - in some cases, the imperial powers even had to hunt down descendants of the sheikhs of defunct tribes and invest them with authority. And a lot of today's radicalism is the product of colonialism and the aftermath, where Westernized elites imposed alien autocracies on the population, not to mention having a colonizing state (Israel) dumped on top of them. Also, the initial governments of the Middle East were often rabidly secularist, and were downright hostile to Islam - and religious persecution has a tendency to radicalize believers, especially if you've undermined the institutions that "kept it real". Most Muslim "fundamentalists" and radicals have no idea what the Sharia really is, for instance. Leo Caesius was just mentioning the other night that Afghans have been demanding Sharia courts, and now that they have them, Sharia judges keep throwing out cases because Sharia law is way more liberal than people think it is (for example, you have to have four witnesses to the actual act of adultery to convict someone of it. It's not very often that people commit adultery in front of four people. Also, women have very significant property rights and civil rights protection in Islamic law.)
The late empire had a parliament, local legislatures, and was in more or less on a course of liberal-democratic development. That's not to say things couldn't have gone south, but I would expect a surviving Ottoman Empire to be more progressive than any country in the Middle East, including Turkey, which has that has that semi-fascist xenophobic paranoia thing to get over.
I'm not sure it could have developed much more liberally than it did in the 19th c - it was probably a bit less so then Imperial Germany and more so than Tsarist Russia. Economically, the late empire was very free-trade, although this was to a large extent imposed on it. The last real Sultan, Abdul Hamid II, halted liberal-democratic political development at the center, but encouraged it in all other fields, i.e. education and in local government. The Young Turk regime started out liberal-democratic in spirit, but drifted towards autocracy as things went seriously south largely due to their disastrously incompetent foreign policy.
I guess in summary, I'm saying the existing structure was a better basis for evolution and development than having it shredded and replaced by multiple alien impositions, or to put it another way, evolution over revolution. It took Europe over 1,000 years to recover from the fall of the Roman Empire - the Middle East isn't going to get over the end of the Ottoman Empire after just 90 years.