“Bone cancer in children?” That's what Stephen Fry said when asked why he doesn’t believe in God. Pointing to evil in the world and asking why God can’t stop it is the most popular way of rejecting Theism today. But it's not new. It has existed for ages.
It's one challenge that leads to another in a domino effect...
1) God is not as knowing or powerful or good as the Theist claims He is. If God is all knowing, then surely He knows all the evil that takes place in the world, and if He is all powerful, surely He has complete ability to remove those evils. If He is all good, then surely He desires that all evil be removed. However, we see evil things in the world all the time, so one of these attributes must not exist, and therefore the very definition of God is a contradiction.
2) If you find a way out of Problem 1, then you are cornered into Fatalism and Quietism: you should submit to evil because it is from God. Trying to change it would be rebellion against His will.
3) The only way out of Fatalism is Deism: that yes God made the world but then left it to us to figure out by ourselves.
4) If He left the world to us, then He no longer has the right to tell us what to do. And that's the beginning of Secularism, which is that you can believe and pray all you want but religion shouldn't get involved in real-life law and society.
1) The attribution of "evil" is a moral claim. There is no thing in the universe called an evil. Evil is a judgement upon moral agents and their actions.* Rather, what people tend to mean when they say "evil" is pain. Great amounts of pain. Now pain is something we can objectively assess and discuss.
Firstly, constant pleasure actually numbs and becomes a source of weakness and pain. So the very absence of pain will actually produce pain. Secondly, There are many wonderful benefits that stem from pain. There is no excaping pain. Without pain, there would be no need for compassion, no sense of hard work and achievement. No achievement means no celebration, and so on. Thirdly, pain is not a thing in the universe either; pain is the absence of Divine mercy. Are we entitled to His mercy at all times? Of course not. In fact, the only way we can even identify His mercy is by having it removed from time to time.
Now if they say, "Okay, I'll accept your claim; evil is a description of moral agents and their actions, like criminals and their crimes. Why would God allow that?" The first answer is that harmful actions are a branch of free-will, which is the greatest gift after life and sanity. (Taking away someone's free-will through imprisonment is universally viewed as harmful and wrong.) But free-will means free-will; it entails the ability to do good, bad, or otherwise. You cannot have free-will and turn around and ask why God didn't interfere when someone does something we don't like. The second answer is that God didn't just leave us to harm ourselves. He sent us Messengers and a Sacred Law for exactly this purpose. But He will not do our work for us or else that would stifle our free-will. We have to voluntarily act upon those laws to stop harm.
2) If reason alone is our criterion, then we would not really have an answer to Fatalism. But we have Revelation, which informs us that the Divine Will is divided into two branches: that which He creates (Irada Kawniyya), and that which please Him (Irada Shar'iyya). Moral agents must make decisions and act solely upon the latter. If Allah creates a thing, then tells us to fight against it, then we are obeying Him and not rebelling. In the same way, a teacher puts wrong choices on a challenging exam, but tells us to fight through them and choose the correct one. Tests are the only way we can know for sure that we've understood the material. Likewise the tests of life are the only way we can know if we truly believe or not. This ends the domino affect nuetralizing challenges 3 & 4.
*In Sharia evil is defined as the disobedience of Allah that is not followed by repentence.