The so called "problem" of evil is one of the oldest arguments against Christianity, and also one of the easiest to refute. There is a simple syllogism to follow that easily demonstrates why God will not simply destroy all evil right now:
Premise 1: The Bible says we are evil.
Premise 2: The Bible says God loves us.
Conclusion: Therefore, God has a good reason not to destroy all evil right now.
But the Bible is a rather large book. We could just end it there, but why would we reduce such a complex subject to such simplicity? So, on top of the syllogism that shows why God has reason to be patient because He doesn't want anyone in Hell (2 Peter 3:9), we can also point out that evil points us to the goodness of God.
The Bible as a whole tells the story of a good God who created a good world, including human beings to bear His image. Humans sinned, and so God cursed the creation, ultimately formulating a plan to redeem His estranged image bearers. All throughout the Bible, we see examples of God's people suffering, and as a result they cry out to God. Ultimate hope comes from the fact that nothing in this life is in vain if it is given to God.
The Psalms especially exemplify this. Aside from multiple pleas for forgiveness, and praises for good things God has done, there are large numbers of cries for deliverance, vengeance and healing. None of these God honouring Psalms would exist, or make sense, if evil did not exist.
Thus, far from disproving the Bible, evil is essential to the Bible's existence. In theory, God could have created Earth as Heaven right off the bat. He could have made it so humans could never sin, and we'd be living in a very different world right now. But in His infinite wisdom, God decided it would glorify Him more if He gave us the freedom to choose between good or evil. Some of God's greatest attributes are the ways in which He responds to evil, be it justice for the impenitent, or grace for the righteous. But these attributes could only be expressed in a world in which evil is possible. Therefore, by allowing evil to exist, giving us cause to cry to Him for the justice only He can provide, or to repent of our own evil to receive the mercies and grace He is most famous for, God is glorified even by that which seeks to disgrace Him. Thus, how can it possibly be used as an argument against Him?