Well, 'become' is a bit of a misleading word in evolutionary contexts. Nothing 'becomes' something else, it simply gives birth to offspring that are a tiny little bit less like itself, and more like something else. The changes add up over time. A LOT of time.
It may or may not help, but some modern scientists classify birds AS reptiles. More to the point, not all dinosaurs were considered birds, but all birds are considered dinosaurs. This is because the definition of what it means to be a 'dinosaur' relies upon the shape of the hip bones, and modern birds share the same shape of hip bone as ancient dinosaurs.
It probably doesn't help that many of the ancient lizards that the layman calls dinosaurs, aren't really dinosaurs at all. Dimorphodon? Not a dinosaur. Pterodactyls and their relatives? Not dinosaurs. Crocodiles et al? Ancient in form, but not dinosaurs. It was only a specific subset of ancient reptiles that were dinosaurs.
Plus of course we know these days that some dinosaurs had feathers; they are preserved in the fossil record. In fact, if you take a bird's feather and a reptile's scale and a bird's feather and put them under a microscope, you'll find they have a very similar structure. The feather is kinda like an elongated, fluffed-out reptile scale.
So, birds are reptiles; feathers are scales; not all dinosaurs were birds, but all birds are dinosaurs. Hope this helps!