I've written plenty of C web apps back in the old cgi-bin days, for no particular reason, and haven't found the memory management an issue. cgi-bin is a great example of a lifetime-managed execution environment where you can just use the heap as one big arena allocator, with malloc() having unchanged semantics (or, if you're feeling fancy, going to a bump allocator), and free() being no-op'd (or, more realistically, just omitted when writing code). Yeah, your high water mark memory usage might be a bit higher than a more managed approach, but the OS is a perfect garbage collector, and the fastest possible garbage collector, for short-lived executables.

The bigger issue is the mediocre string processing libraries that are so common to C.