The first half of the movie, Eggsy (the main character) doesn't seem good in fighting. We don't see him really fight but he certainly doesn't seem that good.
He gets bullied by his father in law even when he is alone with him (when the father is with all his friends, it makes sense not to do anything at 1 vs 5-6).
I understand he might have had some training during the "interview" process but when he goes to save the world, he has become a real machine. And the interview process doesn't last for more than few weeks at best.
So how can he become this good in such a short time?
And also, the last contestant who got eliminated from the interview, doesn't seem like he can fight in the last scenes.