There's an amazing number of users reporting the same issue here with zero contributions from chess.com, which is not very encouraging!
I have had the same issue, and as others have pointed out, it shouldn't be anything to do with system resources being inadequate... presenting the interface for a game of chess just cannot possibly require massive resource loads!
My observation - fan goes berserk with CPU usage up around 40% (compared to barely topping 5% for any other app or website) and stays that way long after shutting Chrome ... and can crash the PC. Task Manager reports that it's specifically Chrome "crashpad" handler (or such) that is gobbling resources, so I figure something on the chess.com site, possibly a 3rd part ad in a frame and/or ad-blocker, is triggering a crash handler which itself crashes creating an endless resource-gobbling recursion.
I'm using Chrome on Linux XFCE but I see others are having the same trouble in Windows so I won't rush to blame Chrome's poor Linux support!
Having this same issue in Chrome on GNU/Linux. Heavily loads the CPU just for a normal game of chess or bughouse, no engine analysis running, nothing.