How far can the sharingan go? The Sharingan makes the user able to follow the target better, which shortens the reaction time of the user. Does the Sharingan make your mind think faster as well?
If it doesn't, the users mind should not be able to keep up with what the Sharingan is able to see. If the mind becomes faster, then the user would become a super genius.

Or is there simply no logic at all?

Naruto has recently learned the flash ability from his father Minato. When someone uses flash are they teleporting or just moving really fast?

Additionally, could any form of the Sharingan "keep up" visually with the "flash ability" of either Naruto or Minato?
Sharingan by no means makes the bearer any smarter. Sharingan enables its bearer to:


See chakra of someone, but not the chakra points
See motions in greater detail making it feels like slow-motioned
Perform genjutsu better
Do sharingan specific jutsu, like Amaterasu, Izanagi, Izanami, etc.


The logic behind it is that normally humans don't use the maximal capacity of their brain, only about 10% if I'm not mistaken. In using the 2nd ability, Sharingan utilize more of the brain to do image processing, but not anything else. As for ability 1, 3 and 4, it's just that Sharingan is like a prerequisite to do so, just like a mere 160 cm basketball player would not be able to do slam dunk unless he have an ultra strong foot muscles. Sharingan is like that, only instead of foot, it's the eyes. Therefore, though Sharingan enables such feats I mentioned above, it doesn't make the user any smarter.

As for Naruto, as far as I know, Naruto is just moving very fast. His fast movement is different than that of Minato. Minato uses Space and Time Jutsu. That means Minato is not moving, he is teleporting. That's why he needs his kunai as marker.

Sharingan would not be able to keep up with it when he's on the teleportation, but when he shows up, it can.