At age 3, that's pretty much where we were, too. Don't lose heart.
This evening, out of the blue, my little boy decided to skip telling me a story, as he often does, and didn't want me to read to him; instead, he just launched into reading to me the 63-page Usborne version of Beowulf, cover to cover. He had read it himself silently earlier in the day. It might be because I video'd him reading a bit of the book, and he then looked at himself. He's seen the videos of himself reading before, and never had this effect.
I don't mean to brag or anything, but I am very proud. He read as quickly and fluently as most people would, with excellent pronunciation, actually looked up the pronunciation of "Leofric" (?) from the list-of-characters page (and subsequently used the correct pronunciation that he learned), used different voices and generally interpreted the text. I could understand almost everything he read without actually looking over his shoulder (which he didn't let me do). Ordinarily, I would have just fallen asleep in this sort of situation, and I was about to drift off in the beginning, but he was reading with total conviction, so he kept me awake. I was very impressed by his pronunciation. I might have corrected him maybe 2-3 times, and he read with correct pronunciation a number of rather advanced words that I didn't think he would know. Once I was explaining the meaning of a word (as I do) and he said, "OK Papa, that's enough." He wanted to keep reading.
He's read more books cover-to-cover in the last month than he read in the entire previous year. I think he's finally becoming an independently-motivated reader...
Now he is asking for another version of the book. We put a few in the Amazon basket but then thought better of it--we'll just check out what's available at the library. He's sleeping with his copy of the book under his arm, like a teddy bear--teddy book?
BTW I'll be "going public" soon with the essay & announcements, so I've removed all the videos of my son from public view. I'll be uploading a compilation video under my real name within a week or two.