It looks that way. But that’s not how it actually works.
Every player must always perform all their actions within the allowed turn time. However, you do not have to wait for the previous animation to complete. You can already play the next card, while an animation is still busy. That way, you can play cards within the turn time, even when the total animation time is longer than the turn time.
On the active player’s screen, you see the cards played as they are dragged to the screen, even when the animation will be delayed because of another still ongoing animation. On the opponent’s screen, you only see what appears to be he opponent hovering over their cards while drawing errors on screen as they play the card, and you only see the animation of the card they played after the previous animation completes. Even when by then the turn timer is already over. You don’t see the card when it’s played, but when the animation process catches up.
Note that random card generation (as well as discover effects) are not shown until animations catch up. So if you want to play a turn with long animations, make sure to only use cards already in your hand, or make sure that card generation is played first so that new cards appear before your turn is over.
Some 8 years ago, a French player took this to the extreme, when he used two computers and two accounts to deliberately set up a board state with seven copies of Prophet Velen, each with huge amounts of spell damage, so that just a single Arcane Missiles would shoot over 25,000 missiles. And he had 9 of them in hand.
He played them all, quickly after each other (there was perhaps two seconds lag in between, the time it took the game server to process the actual effect of shooting that many missiles). But he easily played all before the rope even appeared. However, the total animation time was well over 40 hours (!!!)
https://youtu.be/O22HqSvpAxs?t=1484
Look at the video (I made it start at the point where the setup is done, you can watch from the start if you want to know how he did it). You’ll see that playing those cards took just a few seconds.
I have heard rumors about a hack that can speed up or skip animations. But I have never seen them confirmed.
If this is possible at all, then it would be a violation of the Terms of Service, and players exploiting this hack can (and should) have their account banned.
If you have solid reason to believe a player is indeed skipping animations (such as e.g. playing a card that only appears in their hand when more than 75 seconds animation (even when taking into account that their device might be faster than yours) time has passed, then you can report them. Including a replay link would make it easier for Blizard to investigate.
The proper procedure for reporting suspected cheating is described here:
When your opponent plays, you can click on the card in the upper left corner of the screen to reduce the time it is displayed. This saves about a second of time per card played. The animation of the card effect is still played, though.