We, as Christians and Americans, are incapable of "proving" Islam false or incorrect, there is little we can do. But at the same time Islam teaches that apostasy is a capital offense. Furthermore, in order for Islam to be fixed, it must essentially be destroyed, as the core beliefs are incompatible with a multi religious society and the modern world.
So it is an endless do loop, created by the devil himself, IMHO (with no small amount of forethought, obviously). If there is a fix, it is not apparent to me.
Threats of destruction create more radicalization and more violence. The more you try to force something, the more people resist. That is human nature. But there is another way, also steeped in human nature. Islam should be forced to retreat, not necessarily by force alone. Human nature does not like their medieval tactics and they know it which is why they wish so much to destroy anything western and Christian. They know that it would be much easier for Muslims to become westernized than for Christians to become Muslim. Hence they have to guard their own weakness by repressive measures.
All we have to do is let nature take its own course where we can, inundate them with "sins" of Christianity and temptations of western civilization, not to speak of freedom to speak and worship.
What's blatantly obvious is that all the extreme Islamic countries are extremely backward and inimical to modernism. One reason why Turkey is not (as) backward as other countries predominantly Muslim in population is because it is a secular Muslim state.
So, while we may not convince them that Islam is wrong, we can show them something that is closer to them in their hearts. And, in time, and some gentle arm twisting, the seed of doubt and the virus of being able to do what one really likes will do the rest.