This is really an unfortunate misunderstanding that has pervaded the western teaching of the Thai kick. As mentioned in the question above The Golden Kick really answers much of this. As one can see from this video of Karuhat's kick, there is no "turn over the hip!" at least in the way that it is often overemphasized by western coaches.
This is how I answered a related question, as asked on Reddit:
This is a really bad habit of western Muay Thai, and against high level fighters could be very exploitable. I have a fight in mind between a well-known western fighter (to remain nameless) who was absolutely undressed (vs another western fighter) in the most part due to his "picture perfect" overturn of the kick, leaving him out of position to defend himself (or follow up with strikes). Photos of this fighter for year have been appearing as a kind of Muay Thai porn, as if his kick was so "beautiful", when in fact this overturn is quite far from the real way high level fighters, especially from the Golden Age, usually kick.
That's the first part of the question. The 2nd part is really interesting! There is a LOT of sloppy technique in these 50 kick, 100 kick pad burns. And, at least by my lights, some of that is pretty terrible. Hand position goes to shit, heads float, even by some Thais in Thai gyms. You are always training something, right? But...there is an interesting component of how these kicks just kind of float or pop straight up. Sometimes the padman really will angle the pad down, so you are even just kicking up, into the pad. What's really interesting is how much this violates the westernized "Thai" Roundhouse kick. All you are doing is practicing kicking up sometimes. How can that be good? What's cool is that this is creating a groove for the first part of the Golden Kick. This upward motion is not the complete Thai Kick, which does involve a last second whipping over (but not "turning over the hip" as the west really likes, an internal dynamic of up and whip that isn't included in a lot of these 50 kick speed rounds. I think it's best if you do try to whip that kick a bit, and if the padman doesn't just point the pad downward, (and if your hand positions get correct, and your chin drops down, while we are at it). But, the upward movement itself, the core repetition, which from western eyes might be all wrong, from the Thai side is probably grooving the first movement of the Golden Kick, which is really cool.
This is Karuhat's Kick, for those that haven't seen it. He kicks uniquely, but he does present a really beautiful and in some ways ideal form of the kick, in terms of the rise and the hidden, sudden whip: