Islamic radicalism is so popular, it can even survive Islamic terrorists killing Moslems. The ultimate goal of Islamic radicals is a world run by Islamic clerics. A theocracy, with Moslem clergy in charge. This idea has wide appeal. After all, few devout Moslems could object to the idea of everyone living under Sharia (Islamic law) and supervised by Moslem religious scholars.
But what of Islamic terrorists? In general, the non-violent Islamic radicals consider these fellows misled, overenthusiastic and not really devout. For that, their sacrifice is in vain, and they will be going to hell, not paradise. But for many otherwise peaceful Moslems, the Islamic terrorists are (off the record) heroes if they kill infidels (or rich and powerful Moslems). But the terrorists are fanatic thugs if they kill ordinary Moslems.
Given this situation, the leaders of many Moslem nations have to not only tolerate Islamic radicals, but often enlist them as political allies. Western diplomats usually see the reason for this, but it's often hard to explain it all to the politicians and pundits back home. This gets even murkier when you have to deal with cultural quirks whereby an Islamic radical might sound like he's approving of Islamic terrorism, when he isn't. Arabic is a very difficult language to translate accurately, something that amuses Arabs, but drives Western diplomats and media editors nuts.
All this is also another cause of Islamic terrorism. Moslems are unhappy with their dysfunctional societies, and Islamic terrorism is one extreme solution that gets tried. Most Islamic terrorism is inflicted on other Moslems. It only gets noticed in the West when Westerners are involved. Moslem leaders are more aware of this than are their Western counterparts. Thus Moslem leaders will look after their own interests (and lives) and make deals with any Islamic radicals they can. It's good politics, and plain old self-preservation.