Time does exist as an attribute of space. You have space, you have time. You have time, you have space (or maybe space is an attribute of time). You can’t have one without the other. You can travel forwards in time. We do it all the time. Go fast enough or orbit an extremely strong gravity field, and you can do what most people think of as time travel. Want to see what the world is like 3 million years in the future? That’s actually quite possible. The problem is that after your visit to the future, you can’t go back to where you once were – maybe.
There are theoretical designs of time machines that allow backwards travel in time, but not before the time machine itself was built. If someone actually creates a time machine today, they could move back and forth through time, but not visit anything before today. This may be why we don’t see time travelers. Time Travel is possible, but they can’t visit us in the past.
I have a doubts whether backwards time travel is possible. The universe is a big place and if something can happen it will. You know our big giant particle accelerator in Europe? Particles crash into Earth all of the time with million times more energy than what the LHC can produce. If some form of backwards time travel is possible, we would have already seen it happen.