Keep in mind that Rowling's writing style attempted to being an antique, old-English feeling to the modern lives of witches and wizards and often reflected the disparity between the medieval and the modern. From that perspective, using the word turnstile would be boring and mundane (the root word for "muggles," I suspect).
Based on the root word stile, a turnstile is something with a turning gate you would climb through. But when you reduce the concept philosophically, a turnstile is a stile with some dude standing next to it intoning, "none shall pass!"
In the passage you cite, it isn't a muggle approaching the ticket barrier (a wholly modern contrivance every British citizen would recognize by name), it was Mr. Weasley, a wizard! Befitting the nature of his character, it couldn't be a turnstile, but the more ancient stile, a word many British would also recognize as they exist atop those lovely hedgerow walls all over the place. The British reader would easily understand what was happening, all the while enjoying this warm, fuzzy feeling of nostalgia at the back of their head.
So, in this context stile = turnstile, used for aesthetic purposes to foil the modern ticket barrier with the fantasy/medieval feel of the involved character.
Welcome to the world of creative writing!