If both dragons and dinosaurs were subjected to rules of our world, the Square Cube Law and the tensile strength of calcium carbonate and calcium phosphate would limit a flying dragon's weight on Earth's surface to that of a dog. On a planet or moon with much weaker gravity and much denser atmosphere, a dragon could reach the weight of a horse.
I think a flying dragon could probably get bigger than that, depending on its body plan. The largest pterosaurs (giant azhdarchids like
Quetzalcoatlus) are estimated to have weighed 440–550 pounds (200–250 kg). That being said, the body plans of many (perhaps most) well-known fantasy dragons would be physically incapable of flight in real life, if their bodies could be sustained by physics at all. (On that note, I think most of the really formidable dragons in fiction would probably be severely depowered, crippled, or outright killed by an application of real-world physics; another argument in favor of dinosaurs coming out on top in this scenario.)
Under mythological or fantasy rules any outcome is possible.
That's the problem I have with this kind of match-up. Being fictional creatures, there are no set standards for what dragons look like, how big they get, what their abilities are, etc., so pretty much any scenario where a dragon is pitted against a dinosaur is going to be a thrown fight. If it's Smaug from
The Hobbit or the "queen" from
How To Train Your Dragon against any dinosaur, it's an unfair fight. Not to mention that you have hundreds of species to choose from to represent the dinosaurs' side, not all of which would be particularly suited for fighting a dragon: HTTYD's six-foot-long "Terrible Terror" could be easily crushed by a
Stegosaurus, but would probably flambĂ a little
Epidexipteryx without much resistance.
Basically, dinosaurs are constrained by reality in terms of size, power, and other factors that would help them in a battle. With dragons, the sky's the limit. If you pick the most powerful dragon to go up against the most powerful dinosaur, it's a grossly unfair fight for the dinosaur. If you pick a weak dragon to go up against a powerful dinosaur, you're deliberately tipping the scales in the dinosaur's favor (That's how I see it, anyway). To me, there's really no version of a dinosaurs vs. dragons battle that could be called fair.
Just a thought I had: in a parallel universe, do you suppose dinosaurs and/or dragons ponder over hypothetical battles between humans, like cops vs. firemen, bums vs. hobos, or drunks vs. stoners?

P.S. Sorry, I don't mean to ruin anyone else's fun. Just stating my views on this subject.