I suggest a new "low mastery" queue type
I think it would be great to have a queue where your champion pool is restricted to those which are below lv3 mastery (including free champions which you don't own). The game would be exactly the same as normal's draft, except for the low mastery restriction.
This would allow for people to learn new champions against others also learning new champions. Of course there are bots for the first couple games, but no one wants to play bots enough to really learn a champion.
This would prevent landslide losses in normal's where people are warming up for ranked on their mains, or simply don't feel like ranked at the time, who lane against people who have <20 games on a champion. I think this game mode would allow for normal's to be more consistent (less first time against humans against lv7 mastery matchups) and also give an area of matchmaking to learn a new champion more fluidly.
A possible problem with this is the possibility that someone has no low mastery champions left to choose from after picks and bans. I don't know what to do if this comes up. Possibly the game could up the limit to lv5, then check if they have any available champions from there.