What if your child doesn't want to read independently because you read-aloud to him/her? The Book Doctor has a few suggestions.
Via The Guardian
As you are finding, reading aloud is the best way of getting your child to enjoy books and it will lead your daughter into reading on her own. From listening to the stories you read to her, your daughter will get more and more drawn into the particular experience that books give her. She'll begin to dream and wonder and journey and, once started, she'll want to do it some more. That will be the best spring board she can have for wanting to pick up a book to read on her own.
Don't worry about her not wanting to read any of the book you are reading aloud; keep that book as an experience that is quite separate from her either reading to you or reading to herself. Some children like to pick up the book that is being read to them and race through it on their own so that they can get to the end more quickly. For the majority, however, they want to absorb the story as it is shaped by the voice of the reader and especially the way in which the reader creates the characters by using different voices.
The final solution to getting your daughter to read alone will be to find a separate time and space where you can introduce her to a carefully selected choice of books which you feel she is most likely to like. As you say she's happy with the books you choose to read aloud; the simplest way forward would be to offer her another book by the same author.