As you may sense yourself, part of the problem is to decide just what question we're asking and what would count as an answer.
Start with space. One way of claiming that space is real is to say that it's a thing -- a "substance", as the jargon of philosophy would have it. (In this sense, a substance is something that _has_ properties or qualities and stands in various relations, but isn't a property or a relation. An atom is a substance in this sense; so is an animal.)
The "substantival" view of space-time has its adherents, but also has its detractors. There are many philosophers who don't think space is real if that would imply that it's some kind of thing. Typically such philosophers would say that instead of talking about space as such, we should talk about spatial relations. Things can be between other things, or beside them or above the, or longer than them or a certain distance away from them. What's real, on this view, are the things that stand in the various relations; space isn't something above and beyond all those relational facts.
Obviously we could think of time in much the same way: time isn't a thing or a substance; rather, events (happening, occurrences) have relations such as earlier than, simultaneous with, ten minutes later than... among them. On this sort of view, that's all there is to the notion of time; it's not a thing, let alone a thing that influences more ordinary objects.
Once again, there are plenty of philosophers who prefer this view, and if talk of the reality of time is talk of there being some special _thing_, then lots of non-crazy people don't believe that time is real in that sense. What's real are things of more ordinary sorts (including the not-so-ordinary things of physics) and events of various kinds. Some of the facts about these things are facts about distance, relative location, before-and-after, etc. This ia an attractive thought because it takes away a lot of mystery: if time is a thing, what sort of thing could it possibly be?
Whatever we make of this view, it doesn't exhaust the issue. We can ask: are spatial and temporal relations objective? Or are they conventional? Is it an objective fact that the clock we use to measure time ticks regularly? Or do we _decide_ for various reasons to treat one kind of time-keeping device as the standard rather than another without there being any objective fact about whether it really ticks regularly? This isn't an easy issue, and there's lots of disagreement about it among philosophers.What we see, however, is that even if space and time are matters of relations, people can differ over the objectivity and hence, in an important sense, the _reality_ of the relations.
Your questions about relativity are good ones and they don't have easy answers. As it's often presented, relativity seems to fit best with something like a substance view of space-time. I say "space-time" because in relativity space and time are mathematically related in particularly intimate ways. According to general relativity, space-time doesn't obey the laws of Euclidean geometry, and in fact it departs from the Euclidean to varying degrees and in various ways depending on the background configuration of matter. In fact, on one common way of putting it, space-time tells matter how to move, matter tells space-time how to curve. On this view, space-time acts on matter and matter acts on space-time. If that's right, space-time seems pretty real. (Keep in mind: in relativity, time isn't something distinct from space. What's at issue are questions about space-time.)
The issues here are complicated, but there's no question of a mere leap in logic. General relativity is a complex and subtle account of spatio-temporal matters. On its face, it seems to posit something akin to the electromagnetic field, except it's a field that acts on all matter. Whether that's the best way to look at it is not an easy question, but what seems clear is that the view isn't the result of any simple mistake in reasoning. It's a matter of how best to make sense of a complicated web of theory and fact.