First of all, most of us agree that there is Evil in the world. But far from disproving the existence of God, Evil proves unequivocally that God exists. There can be no Evil without an objective standard of Good to compare it to. In fact, just as darkness is merely the absence of light, Evil is the absence of Good. We’ve all seen the bumper sticker: God is Good. The inverse is true as well – Good[ness] is God. If there were no God, Good (and Evil) would not exist. There would only be pleasant and unpleasant circumstances of varying degrees – “stuff” happens. But Evil and unpleasantness are not synonymous. One infers a much higher order of things beyond the merely subjective.
An earthquake or a hurricane are not Evil. They may cause multiple tragedies but they had no intent or will. True Evil can only be the result of beings capable Evil intent, human beings. Death is not Evil in and of itself. We must all eventually die. That is an unpleasant fact of life. But murder is intrinsically Evil. Murder is the result of Evil intent. Why does God allow it? Perhaps no one fully knows but free will seems to play heavily into the equation. Evil is always the result of man rejecting God.
Finally, The notion that a good and all-powerful God wouldn’t allow any evil or suffering is only a problem if you’re absolutely sure that God couldn’t possibly have any higher purposes for us than to simply make the universe as cushy and comfortable for us as possible.