Suggest using a Caller instead of individual Turns.
Rounds and Turns are still going to be used, at least in D&D, but by using a Caller to communicate with the referee a group gains enormous benefits:
First, no one has down time. Sure, every player engages in the standard practice of communication, listening, then speaking in turn, but that is outside of the game rules. This is down-and-dirty group communication without rules beyond each individual needing to spend the same "in-game" time period as the others via the Caller.
Second, each person gains an advantage by having the others in the group as allies. Planning and working together means players can coordinate their actions instead of bumbling in different directions. By electing to operate as a team actions in the game becomes easier and more likely to end in success.
Third, far more players can be accommodated in the player group. The suggested average of twenty players may be a little much, but not out of bounds by any means. I can only suggest using a Caller as the normal means of play to experience why immediate involvement is available for everyone, while game play actually speeds up. Compared to the 1-on-1, Player-to-GM format it is a hare to a tortoise.
Fourth, independent play becomes the exception, not the norm. Private notes and independently playing the game are still common in my experience with Caller play, but these add to the mystery of each player, PC, and the game. It means there is information your fellow players know, but you may not, and vice versa. These can also add to the level of distrust between players, but group cooperation isn't a requirement for play either. It's just rewarded. Deciding when to share and when not to is a nice feature that doesn't necessarily go away because of the use of Caller.