I'm not sure where you're getting your information on this, but I highly recommend this video of the events. It collects most or perhaps all of the available footage of what happened along with interviews from the main players who were there that night.

After Rittenhouse shot Rosenbaum, he fled the scene and as he ran down the street, numerous people were shouting that he'd shot someone. A few people can be seen chasing after him and when he trips and falls, Huber attacks him with his skateboard and is killed.

Nobody, at least in the footage I saw, says that he's turning himself in.