The "problem of evil" presupposes objective moral values, which requires a transcendent source. So using "evil" as an argument against God presupposes him. Without God, there can be no evil, only a material world governed by undesigned chance or blind fate. So the atheist worldview has the real "problem with evil". If evil is purely subjective, then it really doesn't exist. You cannot make an objective moral judgement on a materialistic universe, even in the face of the most tragic events like the starvation of little children or genocide.
In many instances God reveals a good purpose behind the evil that God ordains. The Bible shows us many clues. (e.g.) The kidnapping and enslavement of Joseph was a direct act of God (Genesis 45:7), yet while Joseph's brothers meant it for evil, "...God meant it for good, to bring it about that many people should be kept alive...". (Genesis 50:20). The most evil act in history was the death of God's own Son, delivered into the hand of wicked men according to the "determinate counsel and foreknowledge of God" (Acts 2:23), for all these did nothing but "what the hand and counsel of God had decreed" (Acts 4:27-28). Yet the good that has come about by the evil act is wondrous indeed, the redemption of poor, deformed sinners, deserving of God's wrath, into adopted sons who have the promise of an inheritance. (Covenant Theology)
"... those eighteen on whom the tower in Siloam fell and killed them: do you think that they were worse offenders than all the others who lived in Jerusalem? No, I tell you; but unless you repent, you will all likewise perish.'"
Related: Is God a Moral Monster?