REST API

My web page has not been maintained for some time. Now I see its become incredibly slow and there is a critical error message regarding site health and performance saying the REST API encoutered an error. I bare understand what a rest api is in general and have no idea what it is on my website specifically, and then how to resolve it?

BTW I know the plugins need update and also the version pf wordpress, but I already tried, together with changing the theme and then first I got the white screen, then the whole website disappeared, so I hesitate to do this now before I resolved the root problem.

The page I need help with: [log in to see the link]