We must first bear in mind that it is not God, but rather sin that is responsible for the suffering in the world. Before man sinned, everything was in a perfect state. Man sinned and as a result death and suffering came into the world.
Romans 5:12 (NIV) Therefore, just as sin entered the world through one man, and death through sin, and in this way death came to all men, because all sinned.
When a person sins (e.g. by killing someone), he inflicts suffering on many people. God is not responsible for causing the suffering; that person is.
God allows suffering because He allows humans to have free will. Free will to do good and to do evil. For God to eliminate suffering totally would mean He has to create robots with no capacity to choose to do evil. And no free will to choose to do good as well. That was never His intention.