First of all, the Flintstones was not a natural history document, it was a highly fanciful and scientifically inaccurate cartoon. Do not look for science facts in entertainment, museums and libraries are far better for that purpose.
As for how birds could coexist with other dinosaurs, the reason is simple: as far as we know, only a single species of dinosaur evolved into all modern birds. Although it is possible there were birdlike flying animals evolved from several lineages of maniraptoran dinosaurs in the Jurassic and the Creatceous period, all surviving birds share the same ancestor.
Furthermore, the maniraptoran dinosaurs are only a small subset within the whole of dinosauria. While the bird-ancestor was evolving into a bird, a tyrannosaur ancestor was evolving into a tyrannosaur, a ceratopsid ancestor was evolving into a ceratopsid and so forth. All the different lineages were going into their separate ways, until all but the bird lineage became extinct at the end of the Cretaceous. Some paleontologists even maintain that certain maniraptoran dinosaurs such as Velociraptor and Oviraptor may actually be early birds that became flightless.
So the statement "dinosaurs evolved into birds" is logically inaccurate if you assume it to mean all dinosaur species. Meanwhile the statement "birds evolved from a dinosaur" is accurate, far more exact, and a preferrable substitute for the former statement.
Source(s):
Paul, Gregory S. (2002). Dinosaurs of the Air: The Evolution and Loss of Flight in Dinosaurs and Birds. Baltimore: Johns Hopkins University Press, 472pp. ISBN 978-0801867637.