There is honestly no easy answer to your question. I would say that the biblical witness is not that God allows innocent people to suffer, but that he suffers with them. In Isaiah 52:12-53:13, the figure of the Suffering Servant, and image used to describe Christ prophetically is depicted as one who bears the sicknesses, infirmities and sufferings of the world as well as their sins for the sake of their deliverance. But he also does this out of solidarity, in order draw people closer to God.
The reality is the world is a screwed up place because Evil exists in it, but Evil is not of God but the antithesis or resistance against God himself. God does not like the world anymore than we do and in fact works actively against these things through the death of Christ, which 1 Corinthians 1 tells us defies the wisdom and powers of this world. Unfortunately, unless everyone should choose to follow God at once, however, the world will not be perfect, and in Romans 8 we are told that the whole world waits in pain for redemption (not just people, but all of creation suffers from the oppression of Evil).
In the last chapters of 1 Corinthians, we are also told that God is calling the whole world to himself and reserving his final work of judgment of Evil so that all might be redeemed. I'll admit, I'm not sure how this last part works, but I hope it means that in the end, the suffering of innocence earns them some eternal reward (though that certainly does not make it any easier to bear in the here and now). The biggest thing is that there is a clear call on those who follow God to fight injustice and suffering of the innocent, and yet so many who would call themselves Christians do nothing.
I would say that the biggest question is why does the Church, which is supposed to be the body of Christ himself, Christ made flesh after his ascension, for the world to see and know, do nothing so often when so much suffering takes place in front of them? I do not think God allows suffering but weeps for it and fights it actively through Christ and obedient Christians, but unfortunately so many of his so-called soldiers on the ground so often fail to follow their orders.
I know this can only begin to scratch the surface of your question. I have faced it myself so often and prayed so much for God to give me the strength and resources to do more. I would say to you, if you want something done, then allow yourself to be God's ambassador and do it.