Avian genomes are small and streamlined compared with those of other amniotes by virtue of having fewer repetitive elements and less non-coding DNA. This condition has been suggested to represent a key adaptation for flight in birds, by reducing the metabolic costs associated with having large genome and cell sizes. However, the evolution of genome architecture in birds, or any other lineage, is difficult to study because genomic information is often absent for long-extinct relatives. Here we use a novel bayesian comparative method to show that bone-cell size correlates well with genome size in extant vertebrates, and hence use this relationship to estimate the genome sizes of 31 species of extinct dinosaur, including several species of extinct birds. Our results indicate that the small genomes typically associated with avian flight evolved in the saurischian dinosaur lineage between 230 and 250 million years ago, long before this lineage gave rise to the first birds. By comparison, ornithischian dinosaurs are inferred to have had much larger genomes, which were probably typical for ancestral Dinosauria. Using comparative genomic data, we estimate that genome-wide interspersed mobile elements, a class of repetitive DNA, comprised 5-12% of the total genome size in the saurischian dinosaur lineage, but was 7-19% of total genome size in ornithischian dinosaurs, suggesting that repetitive elements became less active in the saurischian lineage. These genomic characteristics should be added to the list of attributes previously considered avian but now thought to have arisen in non-avian dinosaurs, such as feathers, pulmonary innovations, and parental care and nesting.