But Chinese (or mandarin) is not a context-free grammar whereas I believe that encoding a language on a turing machine implies a context-free grammar so this example doesn't hold.
Well, a couple of points: its not obvious that Chinese doesn't have a context-free grammar: see the talk by David Branner: "The Grammar of Classical Chinese is Very Close to Being a Context-Free Grammar".
And a properly programmed turing machine can parse languages which are way more complex than context-free languages are.
And a properly programmed turing machine can parse languages which are way more complex than context-free languages are.