Why?
Because individual freedom of choice is worth the cost. If God prevented evil from happening, there would be no evil. But the absence of evil would come at a price: no freedom, and therefore no real good. People would act in an acceptable manner only because they were forced to. Because evil would never happen, no one would comprehend the value of good. God has said that man's purpose of existence is "that they might have joy," and joy only comes from willfully living right. Character is built when a person can selfishly choose evil but persistently chooses good instead. Where there is no freedom, there can be no character.
Why not prevent at least the worst tragedies?
Because God has given each person the potential to become great, each person also can use that same potential to become terrible. The only way to remove the worst is to remove our potential to become greatest.
What could possibly be worth this much suffering and death?
Everlasting world peace. Indescribable, persistent, absolute love. Eternal joy. Forever faithful family and friends, whose character will never diminish. Power to improve the lives of others beyond what we can imagine. Permanent absence of fear, guilt, hate, jealousy, and corruption, and permanent presence of love, kindness, caring, and selflessness. And all of this achieved not by force, but by the willful choices of those who have built such character through persistently choosing the good while here in this training school we call earth.
Because of Christ, we will all be resurrected one day, and our physical bodies will never again be subject to death. Because of Christ, we can choose to rectify our mistakes and learn from our errors during our lifetime, refining our characters into that worthy of the ultimate utopian destination we call the kingdom of heaven.
God won't force us to become like he is so that we can live in perfect joy forever. But he has given us the freedom to choose good or evil so that we can build a God-like character if we desire it. God loves us too much to stop all tragedies from occurring.