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Time stands still

But how is it possible that lag affects only my time (so it passes during those fast moves in the end of the match) and not my opponent's time (so it basically stands still)?? What am I doing wrong or what is my opponent doing to avoid this situation? This is very frustrating
It's simply premoving: Black entered the moves during White's turn, so their move take zero time.

White played pretty slow, black premoved an easy checkmate.

This is perfectly fine. There is nothing you can do to prevent it. But you can learn how to do it yourself.

Premoving comes with high risks, of course. That move will be played if it is legal, no matter how bad it might be.