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Do minion waves really matter?

I've seen in a lot of replays my team just face tanking a lot of turrets while minions are almost two turrets behind in the lane, this leads most of the games to base rushes. Someone knows how to explain this behaviour?
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Are there turrets ? Haha, what i have seen is that only playerstats matters, correct me if im wrong
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They matter, but have a minor impact in most games.
At first: super minions are broken.

The normal minions can be set up, to push turrets on their own after min 15(grouping time).
At this point the real useless things are turrets. They do not even deal damage to minions, kinda.
So when teams just run down mid to double ace each other, the team with the better side lane minion control usually wins. It just takes like 30+ mins sometimes.