Why does God allow suffering?

There's no easy one-size-fits-all answer to the question. And people will ask it for many different reasons - some from merely an academic point of view, others out of their own personal experiences of deep grief or pain.

One reason God has allowed situations of suffering is so that he could demonstrate his power to change them. And as we look at Jesus' life and see the many ways in which he healed people on a physical level, we see a clear anchor upon which to base our faith for how he is working at a heart level.

He that could raise the dead with a word can just as easily raise man from the death of sin. He that could give sight to the blind, hearing to the deaf and speech to the dumb can also make sinners to see the kingdom of God, hear the joyful sound of the gospel, and speak forth the praise of redeeming love. He that could heal leprosy with a touch can heal any disease of heart.
J C Ryle, Holiness
We can trust that the Jesus who healed bodies, can also heal hearts, minds and lives, and graciously use them in service of his kingdom. Jesus' power over visible suffering points us to his power over the greater invisible enemy of our sin.