People don’t believe in God because they suppress the truth by their ungodliness and unrighteousness. Two big words, but both reflect the problem inherent in all humanity. Ungodliness means to simply be un-like-God. We were made in His image, good and pure. In our disobedience to Him, our sin, we wounded this image and fell into darkness. Evil is rebellion against God, it is not simply the absence of good, but the active resistance of good. Humanity became evil.
So why does God allow evil to persist? Because He loves humanity; because He is kind, patient, and good. In His lovingkindness He allows us to keep existing. If He were to eliminate evil from existence then it would mean the destruction of every human being. But He did not do this. He knew there was another way, it was by Him sending His Son Jesus. Before He came we lay dead in sins and trespasses, unable to do anything but evil. Even our righteous works became as filthy rags because God saw the motives of our hearts.
You see if God was simply good and all-powerful then the problem of evil would be no problem at all. He could wipe us all out, get rid of the wicked and corrupt creation. But God is also love, and love is not willing to cast off forever. Love will take blows for the one it cares about. Love will sacrifice everything it is and more for those who are undeserving. Love doesn’t give us what we deserve, it does not keep account of our wrongdoing. And though it doesn’t rejoice in injustice either, yet it suffers long with unjust people. God in His mercy and love sent Jesus to save us from our sins, to remove the corrupt nature within, and to give us a new one, one remade in the image of God, in true righteousness and holiness.
So why does evil exist alongside good? Love. Not that we were deserving of such love, but it is because of how awesome and wonderful God is. It’s just who He is by nature. Adam and Eve chose to rebel, to live in evil, and we each do the same. It is inherent in our nature to do evil, to rebel against the good God. Humanity continues to live in the image of the evil one, of its murderous lying father, that is until the light breaks in upon our darkened hearts and minds, and then (and only then) do we see the good news that Jesus Christ is. He is the loving solution to evil.