Over time the idea that you could cs well and you'll climb is one of the most oversaturated pieces of advice that bargain bin coaching has given us. The problem was that it created two problems. The first is that now, everyone just cses well. Back in s3 when I started, Bronze 5 players averaged 2-3 cs a minute. Now? I see Bronze games where the players that aren't getting dumpstered are getting 6-7+. The other problem is that nobody knows what to do with it. It's great if you're getting ahead of your opponent, but if you're not playing a champion that can destroy the enemy with AoE, you're not gonna go far with a lead.
That's why I say learn good wave manipulation. Wave manipulation is a huge part of this game and it goes almost entirely under-appreciated for what it does. With csing, the best thing you're hoping for is that your team doesn't feed while you get to the point you can pop off.
With good wave knowledge, you can create pressure, and that allows you to
-Force your opponent to lose cs and xp and/or be a free kill for you
-Force your opponent to stay instead of roaming on your lanes
-Force the jungler to camp your lane instead of do anything else productive (if he can't kill you, but your jungler gets a single kill, you've already made a net positive)
-Allow your jungler to counterjungle because you can back him up
-Protect your jungler from invades
-Take dragons while nobody is looking
-Get vision while your opponent is under tower
-Pick up free kills because you don't have to sit in lane as often
Essentially, you can't stop your lanes from feeding their lane opponent in a 1v1 setting, but through pressure you can remove or diminish their chances of dying to external factors significantly.
It's OP. If you want to get started here are some easy tips.
-The best waves to back before are cannon minions. They're harder to kill so even if your opponent pushes hard and you lose the cannon you'll get most of the remaining wave. On non-cannon waves you'll miss 4-5 minions if it gets shoved to tower.
-These are also the best waves to roam during, for the same reason.
-Use the regular waves to disappear from lane and get vision before the cannons come. You can also play mindgames and roam during that time but it's very risky
-The three types of lane positions are slow push, hard push, and freeze. You want to use these to deny as much cs as you can.
-Freezes deny enemies gold the best when you know they're coming back to lane and you either can't shove into tower hard enough or when you can't get anything out of it (likely to get ganked and die). They're also good for staying safe from ganks.
-Hard pushes work best when you can shove into tower to deny minions. These are good for when your opponent is roaming and you cannot follow them (4 gank bot but your jungler is top). Take a plate if you can during this time.
-Slow pushes are when you have 1 more minion than the enemy. During this time another wave will come and since their minions will have done substantially less damage you can have 10+ minions pushed into their tower at once.
-Slow pushing this way often gets you a reset (lane goes back to the middle) or reverse push (their minions come to tower just as yours are dying, giving you 5-6 healthy minions). Slow push into reverse is often one of the best ways to play against assassin lanes.
-These basics are super important for getting kills and snowballing in low elo, and you play around the enemy trying to use them for outplays in high elo.