I'm of the impression that "the universe is getting bigger and bigger" is the other side of the coin from "space time is curved in this really weird way". Also, I'm under the impression that a curved spacetime doesn't contradict the notion of 3D space or 4D spacetime. As I understand it, a flat spacetime is one in which space neither expands nor contracts, and convex or concave spacetime refer to space that either expands or contracts (I forget which is which). Based on observations that things are getting farther apart (or perhaps that they are accelerating away from each other?), we think spacetime has a curve.

I'm sure I'm misunderstanding something; hopefully someone can jump in and correct me.