There're few things in League of Legends that feel worse than helplessness, especially helplessness in the face of what should be an imminent victory. Consider, how many times has your team picked up a few uncontested kills in a close game, only to be unable to use them to accomplish anything because of the hordes of minions that threaten to level your base if you leave them unanswered? If you've played against Riven or Fiora half as many times as I have in recent months, that answer is probably an uncomfortably large number. The truth is, and I apologize if I've hammered this particular point home too many times, League of Legends is a tower defense game wearing the trappings of an action game. Killing the enemy champions doesn't mean anything if it doesn't get you closer to winning the game, and if the minion tide is in favor of the enemy team, that's likely exactly the situation you're going to find yourself in more often than not.
That is, unless you take the proactive approach to minion management. It's easy to realize you need to clear the minions in the bottom lane once the minions are already hitting the turret. By then, though, the damage is done. The tempo of the game is already solidly in favor of the other team, for while you clear bottom lane they're able to move around the map and get the vision, kills, or Baron Nashors that they so strongly covet. Imagine, then, if you were the one who was in that lane thirty seconds soon and who had shoved the creep wave instead of the other team? Too often players return to base, either willingly or not, and simply run to wherever it was they were before, or group up awkwardly with their team. That is simply the wrong play, as the first thing you should look at upon returning to base is the state of your minion waves. Is bottom lane slightly pushing against you? Get down there and turn that push around while you still have time and initiative!
Side lane control isn't only important for when you're winning, however. In fact, it's even more important for when you're not in the lead. If the enemy team is strong enough to win every battle as a group, then don't engage them: engage their minion waves. Split pushing until you've recovered in gold is a very viable strategy in League of Legends, and one that many popular team compositions aren't easily able to handle. The proper answer to a split push is usually to collapse on the three or four man unit in the middle of the map, but that's not nearly as easy as it has been in the past right now. Teams typically are far more disengage-oriented than they are engage-oriented, so it's far easier for that three-man unit to escape than it is for the five men to lock them down. Meanwhile, while the enemy team is fruitlessly chasing kills, your top and mid laner are taking down their inner turrets in the side lanes.
Certain champions even excel at this style of play, and not just the split pushing top laners that instantly come to mind every time split pushing is brought up. In fact, many of the game's best split pushers are actually mid laners. Jax may tear a turret to pieces if he gets there unanswered, but he doesn't have the same easy laning phase that an Ekko has, for example, who will tear apart the turret just the same thanks to Lich Bane. Twisted Fate is the game's first real split pusher, and if you're looking to snowball a lead thanks to asymmetrical fights, no one does it better than the League's classic card slinger. Even some champions that don't itemize into Lich Bane are excellent candidates to hold down a side lane such as Aurelion Sol, whose map mobility is nearly unparalleled and is notoriously difficult to lock down once his core items are complete.
The lesson is simply this: remember that teamfighting is just a means to an end, and going through the enemy team isn't the only way to get at their invaluable structures. Even if you're only pushing the waves in order to force a reaction, and have no intention of split pushing turrets down, you'd still be wise to get your behind in a side lane every opportunity you get so that you might enjoy having initiative more often than not.