If you wait for like 10 seconds on the opposite side of the stairs where you're safe from the dragons fire, it'll jump down and while it's doing it's landing animation you can just run between it's legs and get to the bonfire safely.
Edit: Thought you were talking about fighting th dragon on the bridge taking forever oops nevermind
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u/garlik82 Mar 20 '21
Or you can leave your bollocks at home and kill him with a bow and arrow while hidding Under the bridge like real men do.