The fact that light cannot change speed is one of the core axioms of relativity
That’s light as an aggregate wave. Photons, actual light, always travel at c. What’s happening in a medium is the rapid absorption and readmission of photons. The probability of admission is based on structure of material causing things like lens or mirrors to work.
You can think of it as the photons having to jump between platforms before the can continue running at c.
But doesn’t relativity explicitly state that c is the speed of light in a vacuum, and travelling through other mediums explicitly changes and is explained by relativity?
I am 100% a layman and do not know the answer.
No, they don’t. They can get absorbed and re-emitted, and the space they are moving though can compress sideways. But they can’t make curves at all.