"What time is" is a philosophical question.
And science, in general, can't truly answer philosophical questions.
For instance, science can't really answer the question of whether you exist, whether I exist, whether this post I just wrote exists, or whether bricks exist. Those aren't scientific questions, they are philosphical ones. So, taking your question literally, science can't answer the question of whether time exists or not.
However, some philosophical assumptions make things much easier to teach and understand. If you're trying to understand rigid body mechanics, for instance, it's helpful to believe that bricks exist, even if the philosphers can't quite settle the point.
It's very common to philosophically assume that the space-time continuum exists, and that it is four dimensional. And it makes special relativity much easier to understand. This is rather similar to the way that it's convenient to assume that bricks exist when trying to learn the Newtonian physics of rigid body mechanics.
There are theories, such as some versions of string theory, that suggest that there are more than just four dimensions. There are also theories that suggest that space-time might not be a continuum, such as the "quantum foam" idea. But in the context of special relativity it's convenient to assume space-time exists, and that it's a four dimensional continuum.
If you are actually wondering about why special relativity talks about space-time as a four dimensional continuum, rather than a separate three dimensional continuum plus time, there are some reasons for this point of view. The "parable of the surveyour" gives some insight.
<<link>> has some information about this, (though it's unclear how long the link will be accessible). The textbook reference, which being in print will be around longer, is Taylor & Wheeler's "Spacetime physics", in chapter 1. You can find an older edition of that text
<<here>>.
To give a quick summary of the parable of the surveyor, I would say that it explores the question of why we consider the surface of the Earth to be two dimensional, rather than two one dimensional quantities. I.e., why do we think of north- south distances and east-west distances as being unified into a single conceptual entity, rather than two different things.
Then it draws an analogy between this case and the space-time case, and suggests that the argument is similar. A key point in this argument / analogy is something that is notoriously to get laymen to understand, though - the idea of "the relativity of simultaneity".
Note that much of this is in the context of special realtivity. If you move onto something else, at some later date, like string theory, you might have to revisit these assumptions - in the string theory case, you might have to add more dimensions. Some people have difficulty with keeping all the contexts of the assumptions straight - I can have some sympathy for that, but I don't have a solution.
