The following video about sums up the way most people think of the Time Turners.
Why don't you just use them to travel back and kill Voldemort? Or even Hitler? As Snape says, you'd save countless innocent lives in the future.
I have never held back from criticizing Harry Potter. Actually, I have kind of a hate-crush on the series. Despite my sometimes overzealous criticism of everything about Harry Potter, the function of Time Turners as seen in Prisoner of Azkaban is the best depiction of time travel that I have yet encountered in any fictional medium. It is one of the primary things in the series that I have to say, Rowling did perfectly.
Which is weird, because it's the most obvious point of criticism from everyone else.
Whenever you want to ask yourself the question, "Why don't they just go back in time and ...", stop, and ask yourself instead:
"Why don't they just not go back in time and rescue Buckbeak?"
Maybe that's too abstract at the moment. Let me try something else.
You can go back and try to kill Voldemort. Just get a Time Turner and spin it back a billion times. Go ahead.
Most wizards are terrified of time travel. The effects are supposed to make you go mad. Wizards are cool with literal ghosts living in their rooms, and with dragons in their banks, and with actual fortune telling, and with lifting objects off the ground with magic, but they are absolutely superstitious about time travel.
Why are they superstitious about it? Because weird, looping oddities occur that cause all of the wizards to experiment with time travel to go crazy.
It just didn't work.
It's possible every time wizard to ever exist has gone back to kill Voldemort. But it never works.
It will never work.
You can go back in time to kill Voldemort. But you won't.
I know you won't, despite all your effort, because Voldemort doesn't die until the time of the story.
It's not like your free will is impeded. You can try all you want and do whatever you want to kill Voldemort. No mystic force is going to tug at your insides and prevent you from pulling the trigger. If you think about it, there is nothing impeding my free will when it comes to wining the Olympics. But I'm never going to. And you're never going to kill Voldemort.
Its physically possible to kill him. Your bullets won't bounce harmlessly off of his chest, or stop in mid-air right in front of him, or whatever. Not like you'll get far enough to find out, though; if your bullets were about to hit Voldemort, then they would hit Voldemort, and he would die. And Voldemort doesn't die at that time. So your bullets won't get close enough to prove they could hit him.
You're not going to kill Voldemort, because you're just not going to. There's no better explanation. You just don't.
If you did, he wouldn't be the antagonist of the book series. And yet there he is, Avada Kedavra-ing people all over the place. So whatever plans you make to kill him in the past, they just aren't going to work (and may just exacerbate the problem), and you should move on and try something else.
Please note how the causality works here. I am not saying
Voldemort is alive in 1994 ----> You can't kill him in the past.I am instead saying
You don't kill Voldemort in the past ----> Voldemort is alive in 1994.
So back to the original question:
"Why don't they just not go back in time to save Buckbeack?"
Think of what it entails that they do go back to save Buckbeak. During most of the events leading up to the Shrieking Shack, there are two copies of Harry and Hermione. One copy follows the other, and the two interact to a substantial extent. We don't see the second copy because they stay hidden, but we do observe their actions as unexplained anomalies that occur to the primary copy.
By the time the group gets to the hospital room and Dumbledore advises them to use the Time Turner to save Buckbeak and Sirius, those events have already happened.
This is important. The copy of Harry and Hermione seen by the protagonists when they travel back in time is not some fictional, hypothetical alternate timeline Harry and Hermione. They are seeing Harry and Hermione. That is, they are seeing themselves. Just a few hours ago. We just read about these parts. These are the Harry and Hermione. There are no Timeline A and Timeline B versions of the two; there's just Harry and Hermione, and the things that they cause to occur have already, in fact, occurred.
We saw these events forward, then reversed, then saw them again from a different perspective. We did not switch tapes.
And here we have a situation that speaks to the reverse of the usual time-travel paradoxes. The most famous, namely the grandfather paradox, asks what would happen if you went back in time and killed your grandfather (or yourself), thus ending your existence before you can go back in time to end your existence. Another, related paradox asks what happens if you go back in time to kill Hitler, or save the Titanic, or whatever; if you succeed, then you erase your motivation for time traveling in the first place, so you would never time travel, which means you would never make the change, which means you would time travel.
This time we ask a weirder question, sometimes called an anti-paradox: what if Harry doesn't travel back in time, and therefore doesn't save himself from the Dementors?
Clearly, by the time we get to the part where Dumbledore recommends time travel, Harry doesn't have a lot of options. If he decides not to, then he must already be dead. But Harry isn't dead. So Harry is going to decide to travel back in time.
However, it isn't just limited to Harry. If Harry decides not to travel back in time, then Buckbeak must be already dead. But Buckbeak isn't dead. So Harry is going to decide to travel back in time.
And it isn't limited to who's alive and who isn't. If Harry decides not to travel back in time, then those rocks in Hagrid's garden don't get thrown. But the rocks in Hagrid's garden do get thrown. If Harry doesn't, the blades of grass he stepped on won't get pushed down in quite the same way, but they do get pushed down in quite that way. If Harry doesn't, the molecules of water in the lake won't get stirred up like they do, but they do get stirred up like they do. The oxygen molecules he breathed wouldn't have been turned in to carbon dioxide, but they were turned in to carbon dioxide. The entropy of Scotland wouldn't have seen any contribution from Harry's metabolism, but entropy did increase from Harry's metabolism. On and on.
Every single infinitesimal impact Harry had on the universe would not have occurred, and they all did occur. Maybe humans didn't notice each little thing like entropy and O2 --> C02 reactions, but physics is notoriously unconcerned with human cognizance. Those things did happen, and in fact we "saw" them happen on screen the first time through, before Harry and Hermione decide to go back in time.
So that's why they don't use the time turner to stop Voldemort. Because everyone to use a time turner to stop him fails, has failed, and will fail. It's a plan that never works and will never work. There isn't even any reason to try, and anyone who has tried to do anything with time travel has gone insane or worse.
As to why our heros don't at least attempt it, maybe it's because if they did, then the story would be about their colossal failure and collapse into madness, which isn't the sort of thing that tends to make it in to books. At least not children's books.
So, despite everything else I've said on the subject, when it comes to time travel, I really have to tip my hat to Rowling. She did it very well.
Of course, the real question is why wizards would take literally the most dangerous item they know of -- so terrifying that people accustomed to conversing with ghosts leave them locked in a special, secret vault deep underground where no one can find them -- and lend it to a 13-year-old girl so she can get to her classes on time. But that's another post.