Hi GLaDOS (:D),
First of all, you don't need mods, you can make a megafactory with vanilla as much as anything else.
The real thing is scaling. at the beginning, you need little of everything. But as you grow, you'll notice that a little up in one ot your final products output translates into a very consequent upgrade of your raw material needs.
Tbh, you can "finish the game" with a very reasonable factory. Even if one can argue about what "finish the game" is in a sandbox like Factorio.
With Factorio, you're only limited by the hardware. Whatever you do, you can do it faster. You can feed 4 science labs for you research ? ok you can research faster if you manage to feed 8 labs. But for that, you'll need a way bigger smelting zone. and that's the same thing with about anything else. whatever you build, you can build more and faster but you need to scale you input.
And that's how megafactories are born
when you have like 16 science labs running full speed, and need a massive capsule building facility because your pollution has pissed aliens a lot and they have these huge bases, and you have to expand because the tiny ore patches you have at starting point won't be enough, and so on.
What's fun is that you can always make faster, stronger, better. Just aiming for the rocket launch with a small enough factory not to use up all your starting ressources is OK for the few first games, but not forever.
The difficulty often arrives when you try to scale and say to yourself "darn, should have left more space for that". Usually, rebuilding from scratch a little further is a very valid option.
[Edit] I see you're doing a lot of things manually (inserting your sciencs packs into the labs, for example).
The idea of Factorio is to automate. Bring your science packs to the labs with a belt, and insert them into the labs with inserters. and so on ... whatever you do "by hand", try to automate it.