Izuku "What makes you think I'm on drugs?"

If there was one thing people thought about Izuku other than his quirklessness, is that he isn't petty. After all, Izuku acts pretty innocently. So when All Might had said that he couldn't be a hero, Izuku decided that he'll be petty for the first time in his life. Finding a place to build up muscle was easy enough and in ten months he shot up from five feet five inches tall to ten feet. With his body being more muscular than the villain Muscular himself

Of course, he wasn't expecting to be mistaken for a hero when he went to the entrance exam for U.A but he got through it. Though he was a little confused when they asked him for a blood test but gave it anyway thinking it was a normal precaution for the school. Later when talking about the examinee Izuku Midoriya, who they had suspected to be taking an insane amount of steroids, Recovery Girl says that is actually not the case. Recovery Girl then says that whoever tested for Izuku's quirk is an idiot before stating that Izuku's quirk allows him to build muscle fibers and stockpile them. Which is evident since Izuku shrank from ten feet tall to eight and a half feet tall after the U.A entrance exam.

Basically Izuku was misdiagnosed as quirkless. Izuku's quirk here is called muscle stockpile and allows him to build up and stockpile muscle fibers for use. The muscle fibers Izuku builds hold a lot more strength than regular ones and it raises his speed, strength, and durability depending on how much muscle fibers he has stockpiled. He'll also shrink when he uses them.