My suggestions aren't classics, but more recent modern books. My DD also loved the Winnie-The-Pooh books, but then we drew a blank. I tried Little House but she lost interest.
What DD LOVES right now (she is also 4), is the Magic Treehouse series. We read 1-2 each week, and they are short enough you could even read the whole book in one sitting (we usually read it over two days).
I bought a few others before we went to France this spring to read on the plane, and they were also a success. She loved Bad Kitty Gets a Bath...there are a ton of illustrations which I think makes it an easier chapter book for kids this age. We also read Lulu and the Brontosaurus and DD really enjoyed it.
Crazily enough, her current *favorite* chapter book came as a complete surprise for me...The Invention of Hugo Cabret. I actually checked it out of the library for myself, but she started looking at all of the illustrations and asked me to read it to her. We read the whole 500-page book in 3 days. I can't tell you how much I loved it...the illustrations are beautiful and actually tell a part of the story. The smaller amount of text makes it a quick read. There is a chase scene, and one scene where someone is found dead, but it is not graphic in any way, and dealt in the story in a way that I think is okay for kids this age to read. There is also a fire, that is quite sad. We talked a lot about that scene after we read it, but again...the illustrations are not graphic. I can't wait to read his next book that is written in a similar style, Wonderstruck.
When we went to Paris, the one museum we went to was in an old train station, DD thought she was in Hugo's train station. The magical delight she experienced at what would have otherwise been a rather boring museum for her (impressionists) is one of my fondest memories from our trip.