My DD is 6, and we are making the transition from me reading her picture books to sharing chapter books (she's not yet confident enough to read them independently). I continued a diet of picture books, but began to intersperse them with longer texts and less pictures, and there are lots of these sort available. I would agree with the recommendations given above, and also recommend these books as being gentle ways of bridging that gap:
Noddy books by Enid Blyton (I find little in the way of racism or sexism to object to, but you will know where you stand on the thorny issue of Enid Blyton)
Milly Molly Mandy stories - I love the old-fashioned language as I think it exposes children to storybook language that they might not hear in everyday speech
The Mrs Pepperpot stories
Enormous Crocodile by Roald Dahl and The Twits (I personally found some of the language used to describe old people a bit near the knuckle and actually more offensive than Enid Blyton, but again, a personal taste thing)
The Little Wooden Horse - Ursula Williams
The Giant Jam Sandwich - John Vernon Lord. A little gem. It's a short story told in verse, but I think a bright 3/4 y o would love it. Possibly more of a picture book than a chapter book.
Winnie the Witch - first enjoy the picture books (Korky Paul's illustrations fascinated my DD), and then move on to the collections of short stories. You could also try the Worst Witch stories by Jill Murphy, though these have less illustrations and are set in a boarding school, which may be an alien concept to a 3 yo?
You could try as well the Lighthouse Keeper books by Ronda and David Armitage, Mr Majeika by Humphrey Carpenter, the Katie Morag series, The Polar Express, The Jolly Postman, Clarice Bean...oh, the list is endless really. And I haven't even mentioned poetry!