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That's the way it's often related it to people, but a closer look reveals that the actual measurements didn't measure space distortion. What really happened is they used stuff in space ( a probe ) to look at how other stuff in space ( large rotating masses ) relate to yet other stuff in space ( light ) from other stuff in space ( distant stars ). From that, the assumption is made that because all that stuff in space behaves like space is distorted, that space must therefore be distorted. 

But that's jumping to a conclusion that could be completely wrong. For example maybe space is completely uniform and stuff in space behaves the way it does because it's governed by the rules of some vast construct that tells it how to behave according to its own rules ( e.g. the fundamental forces of nature ), and we humans just happened upon an analogy ( curved space ) that can be used to make accurate predictions.

So basically, I say so what if the effect on stuff in space that is associated with massive rotating objects in space can be calculated by drawing lines that look like a ball spinning in a liquid that's a little bit sticky? That still doesn't prove space itself is curved or sticky. To quote the ESA site: " Gravity is best explained using Albert Einstein's Theory of General Relativity, which is not a quantum theory. Instead, it imagines that gravity is generated when matter distorts space, like a heavy object would stretch a rubber sheet."

Notice the words "imagines" and  "like". It's clearly an analogy. But it works in practical applications, so let's keep using it where it's helpful. Just don't fall prey to the logical fallacy of assuming analogies equal realities, because somewhere the analogy will break down, and you'll run the risk of thinking other predictions are also possible in the real world, ( when they're not  ), and I think that's the territory we're getting into when we start talking about warp bubbles and time travel.