Coming-of-Age
Broadly speaking, the genre of Oliver Twist is a novel... but we're probably not going to win any lit studies gold stars for that statement. Slightly more specifically, it’s a serial novel (meaning that it was published in pieces, and can be broken down into shorter episodes). Still more specifically, it’s a "progress" (check out the "What’s Up with the Title?" for more on those), which is just a fancier way of saying that it’s a coming-of-age story.
This is pretty obvious, really—Oliver’s a little kid, and he has to get over various obstacles in order to pass safely into adolescence. But it diverges from traditional coming-of-age stories in that Oliver is still a child at the end of the novel. He hasn’t actually "come of age" by the time we leave him at the end of the last chapter.