Are space and time really connected?

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If so, why does time behave so differently from space?

If we treat relativity seriously then yes, they both create spacetime, so they must be connected somehow. But we shouldn't treat time as 4th dimension, it's rather 1st (0th) dimension. With time we can go only one way, with 1st (space) dimension we can go 2 ways, with second dimension 4, etc.

In Euclidean space, you can swap around dimensions arbitrarily through rotation, but this is not the case with spacetime through Lorentz transformations.

https://en.wikipedia.org/wiki/Light_cone