Not only did time not exist before the big bang, space didn’t either.
This is at heart a metaphysical question. While observational evidence compels us to accept that the universe emerged from a big bang according to details calculated back to the first split-second, the evidence is equally mute on the nature of the singularity itself that spawned this big bang. It’s a deep scientific mystery and will likely remain so in the foreseeable future. What else can we do?
String and brane theories in higher spatial dimensions, compactified to 4D space-time (or 3D in the holographic view), are purely speculative. Absent testable inferences “it ain’t science.”
That doesn’t mean string theorists are wasting their time, however, if they can coax from their models some predictions that might be testable with reasonable technology. (Not, say, requiring a particle accelerator the size of the solar system.) I think there has been some progress along these lines, at least ruling out certain classes of theory.
In one view the big bang & resulting observable universe is but one cell of a higher-dimensional fractal structure of infinite extent, where big bangs routinely occur wherever branes collide. Or something like that.
That still leaves the question of why there is something rather than nothing. More metaphysics.