Is it actually possible to mess up someone’s leg by kicking them in the side of the knee, which seems to be really common in movies? If not, what would be a plausible, effective way to attack/disable the legs?
Yes, absolutely. Any joint you have that you enjoy using that bends in one direction, applying pressure in any other direction until it pops is going to be unhappiness. This is what joint locks do, and joint breaks are simply applying enough pressure or force to break it. It’s about leverage, not strength. Anyone can do it.
The common kicks for this are the sidekick and the shin kick, but you can also break the knee with your elbow, your hand, or a car door. Understand though, once the joint is blown, it takes major surgery to get it back if you can get the joint functioning again at all.
In a real life context, whatever you need to stay alive and escape. They can’t chase you without functioning legs. In a fictional context, you probably want to take this reality into account. This is a joint break, and the knee is a necessary part of a human being’s ability to move while on their feet. Blown joints are usually permanent, or have a long recovery time with modern medicine.
You’re going to want to take that into account according to your character’s own views on violence and it’s uses. The thematic aspects of violence in fiction are as important as the practical applications. Your character’s morals mesh with their approaches, regardless of what is or isn’t best or smart. Everything your character does says something about them, and if your character is one of the peaceable “Everyone must live!” types then a joint break in application creates implied hypocrisy and dysfunction. Catherine of Russia didn’t kill her political rivals, but she locked them up in prison, had them tortured, and this included children. So… what is benevolence? Breaking someone’s arm forever isn’t murder, but it’s also not a nice thing to do and weighing the morality of your character’s actions is something you should consider. One might consider locking a child up in a tower, away from their parents and the sun, refusing to allow them to learn to speak, read, or write, better than killing them. (That was the ultimate fate of Ivan VI.) Some might not.
The question is not just does the approach work, but does the action and its consequences fit with the character’s stance? The second is sometimes much harder to answer than the first, but, for me, the real problem is character actions not matching mentality and intention. Remember, “Does it work?” should always be followed with “Should this character do it?”