In Dutch the main verb is usually at the end of the sentence, so 'lezen' should be at the end. The other way around does not sound natural, but is not wrong.
"Hij wil zijn boek kunnen lezen." sounds much more natural to me.
However, I believe this might be different in Flemish.
These sentences are starting to not make sense anymore. "He wants his book to be able to read" Immediately came to mind but was wrong.
I think it's just a matter of realising that the Dutch and English word order are different, so you can't directly translate the sentences in the same word-for-word order. And maybe in the end that will make it easier to learn the language. My father was Dutch and moved overseas -- and he said he got to a point with his English where he wasn't progressing, and someone asked him whether he thought in English or in Dutch, and he said Dutch. And they told him he had to start thinking in English, and that made all the difference. So, I'm thinking that with the different word order, you start to look at the sentence as a whole and extract the meaning without doing that word by word breakdown, understanding it holistically. I'm not at that stage yet either, but I'm hoping one day it will come.
NicHciN, are you saying that your proposed sentence makes more sense than the actual correct translation from DL?
In any case, Tracey843948 is right: Dutch word order is different, as is explained in the Tips and Notes that go along with this lesson ...