I don't think taking turns would work, it would slow down the whole thing even further.
Rather, each team should have a number of games per round equal to the number of players on their team; each player is assigned to a game, and plays the whole thing out.
Ideally though, the idea of teams would be scrapped all together. This would be best as an individual effort. Right now it's like a team PvP. I think it would work better as battle league, but with connect 4.