Sieging towers is pretty simple, but you need to be careful about it. When both teams are alive and your team is trying to take a tower, it's your job to do as much damage as you can to the tower without dying. You need to do as much damage as you can while remaining safe. This means you can't just walk up to the tower and stand there auto-attacking until it dies; this leaves you open to get engaged upon, unless there's no resistance from the enemy team. Instead you have to repeatedly poke the tower. Walk up to it, attack it a bit, back off if someone dangerous is walking up to you. Now you might think "but what if they never stop protecting their tower? Doesn't that mean it's never safe for me to auto-attack it?". If they're constantly standing in a defensive position in relation to their tower, then they're most likely to be in range for your team to poke them and force them back. They need to be in a place where it's safe for them to sit and look for a good engage, where your team has no way to attack them without tower diving them. And if they advance, that takes them further out of safety, so if you match their movements to stay safe from an engage, they'll be forced to move back for their own safety again, as they don't want to get engaged on in a bad position either, nor poked down to a point where they need to recall to recover their HP.



So you should think of sieging as slowly whittling down a tower. It almost never takes a single minion wave to destroy a tower (unless you have baron buff), it can take many, many waves, but slowly but surely, the tower will go down if you can siege it properly.



When you're attacking a tower with enemies nearby you always want to be positioned at max distance from the tower while you auto-attack, and in between auto-attacks you should move backwards then forwards, so that while your auto-attack is "on cooldown", you're not spending any more time closer to the danger than necessary. It's also good habit in case they start to try an engage and you're already automatically moving backwards the moment you finish your auto-attack, meaning you'll often protect yourself from an engage faster than you can actually react to it, so long as you notice in time to not move back towards the tower for your next auto-attack.