As you can see from one of my previous answers, photons do not have a proper time, but you can use a λ affine parameter, that increases monotonically along the lightlike worldline.
Now you are saying that a photon will travel faster in space then in a medium. In reality, photons always travel at speed c in vacuum, when measured locally.
Now when a photon enters a medium, it still travels in vacuum between the atoms/molecules, at speed c.
It is the wavefront of the whole EM wave, that is built up by a herd of photons in a coherent way, that slows down in the medium, due to the interactions of the individual photons with the atoms/molecules in the media. That interaction takes time.
The simplest picture is that light always travels at the speed of light. But in a material it travels at the speed of light until it hits an atom. It is then absorbed and re-emitted in the same direction, which takes a small amount of time.
The more this happens, the slower the effective average speed.
The denser the material, the more atoms there are in the way.
As the medium gets denser, the number of interactions will be more, it will take more time from the individual photons and the wavefront will slow down more.