I have written a web site in C about twenty years ago, and it was terrible. Absolutely Turing Tarpit category, the language is just completely inadequate for this problem and so you spend all your effort trying to build something which is suitable out of the language, which again, yes, Turing Completeness, of course it's technically possible but it's a bad idea.
Web sites are full of strings, and C doesn't really grok strings. Not in the way WUFFS doesn't grok strings (there are no strings in WUFFS, it has no string type, it's not a general purpose language it's not for that), more like the way New Forest ponies don't grok traffic laws.
Many high-level languages with good string support are implemented in C. The C standard library sucks but it's not your only option for handling strings.
> I wouldn't write a website in C.
With appropriate libraries, why not? The safety issues might be a problem, but you could sandbox it.
> I wouldn't write a robot arm driver in PHP.
I would. It's general-purpose enough to speak binary protocols without too much hassle, though it'd be more fun to use something else.