I played a campaign where one of the casters was able to make his weapon to hit as if it is two sizes bigger than it really is using some spell so I wanted to create non-caster who will be able to do something similar. So now I have an opportunity to create a character of 7 level or lower and started looking for solutions and I did not find anything except "Do a bugbear", but I really don't feel like playing as one. Are there other ways?
You can't get the crunch, but you can get the fluff.
There really isn't anything in RAW that does exactly what you're asking for. 4e gates its effects by tier pretty hard, and heroic tier basically doesn't get size shenanigans. There are a few Epic Destinies that give you ways to be larger size, but that's a ways off for you. Anything you could get to do that at paragon would probably be a 1/day power of limited duration.
That being said, there are ways to get what you specifically want without breaking the game, and thus ways that you might be able to convince your DM to agree to. If what you want is the feel of having your character walk around with a ridiculously oversized sword, then talk with your DM about starting with a custom magic item. Call it "Sword of the JRPG protagonist" or whatever, figure out what the in-game effects are (some sort of damage increase, say) and make sure that it's balanced against the other items that are available.
If you can come up with an existing magic weapon that will give you the crunch to make you feel the fluff, that's even better. In 4e, the DM is specifically encouraged to let you refluff things more or less how you want them to (as long as it doesn't break the fiction). You could even make it a combo item. There's a set of bracers that gives an item bonus to damage. Refluff them as having an enchantment that lets you wield your enormous sword, get a magical sword with stats that make the whole thing feel right to you once you add the damage bonus from the bracers, and you're good to go.