You can start a new Clan within a faction, but starting a new faction freely would make things like War, faction events, wild fights, etc., a bit confused. Plus a stance on ‘cyberbullying’ isn’t necessarily related to the faction story, and I’m not sure that kicking people out of factions is typically possible as a role for one member (although someone with higher BR joining would become leader anyway, regardless of their personal habits.)
However, you can switch faction if you want, ie. between Supremacy and Amity. I haven’t done it, though, so you might want to ask someone else who has. I think that it usually requires around 1k ambers (I don’t use those, but I wouldn’t switch anyway), you go to the Faction window and then to the tab named after the faction you’re not in, then you click Join.
Would also be useful to know your server or name, since otherwise people might not know what kind of things you’re talking about.