Observation is the main tool of science. Observation, replication are the heart of experimentation.
And yes, you cannot know as proven fact, in the world of science, what you can't observe directly.
All you can do is provide evidence of that you think supports the big bang. You cannot prove it empirically.
Nobody has ever seen an atom directly, with or without a microscope. Nevertheless, i think we all agree they exist.
And it has been observated that the universe is expanding, so if something expands, it had to start somewhere, doesnt matter if it is 5000 or 13.7 billion years ago.
but because we have observated that the universe is billions of lightyears in seize, and light has a limited speed, it is impossible the that light has traveled billions of lightyears towards us in just 5000 years.
Not that that is the only evidence:
By the way in the begin post i saw the mentioning of a quasar. 5000 years is totally NOT enough time to even enough time to form an apropirate star for it, not even speaking of the quasar itself.
Also look at the fact that the fotons of the quasar had to travel towards us with the limited light speed. And the quasar happened much more than 5000 lightyears away from us.
I can continue with this for a very long time... I know much about astronomie and physics.