One of the main problems with this hobby are the "promises" provided by marketeers and advertisers...yes, you can make beer in 3 weeks...but is it good beer? That really depends on the ingredients, style, your brewing/fermenting, conditioning and temperature processes.
The 1-2-3 Rule is not really any written rule, it's more of a "recommendation". A starting point. Something that you have to learn to tweak for your conditions.
All beers do not ferment at the same time. There are a lot of variables to know about and try to adhere to to make a really good beer that you would be happy (but sadly) to share with your friends.
Your beer will not be as clear as anothers beer even if you brewed the same recipe.
Your equipment and ingredients may be the same, but your water is different from theirs, your process is different from theirs, your fermenting temps may be different, etc. What comes into play is knowledge of the ingredients; what they do under different condititions, and experience (knowing what to change in order to effect the outcome).
Fermentation is a natural process that cannot be altered without consequences. The 1-2-3 method may work for some people, but not all.
New brewers usually have minimal equipment and are always too anxious, yes, I was too...before I learned that time is your friend when to comes to brewing beer.
When someone (such as myself) has loads of equipment we learn to be more patient with our brews. We give them all the time they need to ferment, clear, and age before we bottle or keg. We build up a long pipeline...and wait...
What you need to do is get your supplies together (kits, malts, hops, yeast, recipes, more bottles, etc.) to brew another batch right after you bottle so you can fill up your pipeline (after bottling). Then you need to let them age a bit. Even waiting 3-5 days longer will improve your brews. Sure, you can sample one or two along the way, but waiting is always better.
You need to make sure your brew is done fermenting before bottling. You need to make sure it's not over primed. You need to wait before sampling too early. You need to place them in the fridge (I say at least 3 days to get acclimated and cold conditioned) before sampling cold.
There are a lot of things to learn about to brew successfully, but one of the hardest and simplest tip is learning to wait...
I think your brew is too young. What temperature are your bottles being stored at while conditioning? If it's below 70F then it'll take longer than the 1-2 weeks you were "expecting" them to be carbonated. It may take 3-4 weeks instead.
But, that's only my opinion...Your mileage may vary....