Bent Time

Here's something that's bothered me for a while, not in the 'keep you up at night' kind of way, but in the 'stub your toe' kind of way. That's the kind of bothered where you don't really notice it until you think about it, and then it bothers you constantly for a little while then goes away. And then you'll think about it again and it'll get you stirred up all over again.
   Why hasn't anyone explained to me the 'bent time' part of bent spacetime?
   Spacetime is the prediction of Einstein's theory of general relativity, which states that gravity is a byproduct of massive objects bending spacetime. All of the examples I've ever read about or seen on TV discuss gravitational lensing, either from galaxies or our own sun, and as far as I can tell these examples only show bent space. That is, you can see multiples of the same star appearing multiple times around a distant galaxy (lensing), or you can see how our sun bends the light from stars behind it, making them appear to change positions in the sky. On TV shows they always roll a ball across a net to show the reticulated distortions in the square grid, using that as an analogy of the sun making a dent in spacetime that the earth has fallen into. Space.
   But what about bent time? I never hear explanations of how time is also bent.
   If massive objects can bend light in space, and if spacetime is the combination of 3 dimensions of space and 1 of time, then shouldn't a bend in spacetime not only affect position (x, y, and z axes), but shouldn't it also affect time? And what does that mean? If our sun distorts the apparent position of a star behind it, then does that mean we see that light before or after we otherwise would if the sun weren't there? And what are the implications for that? We already know using atomic clocks that relativistic effects are measurable for human beings, how does bent time affect us?
   I don't have any answers on this one, only questions. I think physicists are short-changing us when they explain general relativity if they leave bent time out of 'bent spacetime.'
   This is the long way around to saying 'I want my time machine, you bastards.'