I take it you don't mean picture books, but books full of text (and the odd picture), like an adult book?
I'm in Germany, where they don't start to read until 6 or 7, and even then are not expected to read that much. Books tend to be quite babyish or dull until about age 10. (Adults still end up bookworms, however!)
I also tried to teach the kids to read in English at home. My daughter was very keen, read picture books with ease and was reading adult-like books independently by about age 9. My son was absolutely not keen on reading, so I gave up and he learned in German at 7. By age 10 the skill had magically transferred to English and he started reading independently, for pleasure. Now they are 20 and 22, and both read regularly, more than me probably. Their dad almost never reads anything.
I read them both bedtime stories until our son, the youngest, was 10, and then at some point I couldn't be bothered, and left off halfway through a book, and eventually he wanted to know what happened and finished it. After that I made a point of getting things that were their taste, not mine - plenty of easy reads. My daughter had a subscription to "Shout" magazine (awful teenie nonsense!), my son loved a series about teen spies. We also used to listen to audiobooks on long car journeys - e.g. Sophie Kinsella or Michael McIntyre's autobiography, very light - and my daughter worked her way through those on paper, too. I agree that it just has to be fun at that age. They both read more demanding stuff now.