Harry didn't like to be a burden, he didn't want to give the nice adult extra work. Vernon always said he annoyed everyone when he acted so helpless, he had to grow up fast and stop being a burden. So he tried to get off the bike himself, attempting to copy Sirius because he just slid off the bike all cool like.

And Harry tried to do that in Sirius' all too big coat and, hence, no arms to stabilise himself.

He just straight up fell off the bike and onto the floor. Ow. He tumbled. Looking concerned, he looked up at Sirius with panic. "I'm sorry!" Did he get his coat dirty? Oh no. His dad probably never fell over like this. He was being bad.