I'm pretty sure a human language (e.g. English) is powerful enough to simulate a Turing machine, which would make it Turing complete. However, that would imply natural languages are no more or less expressive than programming languages, which seems questionable.
Is natural language Turing complete?
First of all "Is language X Turing complete" is only a well-defined question given a well-defined semantics for language X. It is nearly impossible to define one for natural languages due to natural languages' complex nature and reliance on context and intuition. Most (all?) natural languages don't even have a well-defined syntax.
That aside, your main confusion is based on the assumption that it's not possible for a computational model to be strictly more powerful than a Turing machine, i.e. be able to simulate a Turing machine, but also to express computations that a Turing machine can not. This is not true. For example we can extend Turing machines with oracles and we get a computational model that's strictly more powerful than plain Turing machines.
In the same vein we could define a programming language MagicLang that can do everything an ordinary programming language can do plus solve the halting problem. Defining a semantics for such a language is easy: just take the semantics of the language we used as a basis and add a function bool halts(string src, string input) with the semantics "returns true if the program described by the source code src successfully terminates after a finite amount of time when given the input input". So that's easy. What's hard, or rather impossible, is implementing this language.
Now one may argue that natural language can also describe the halting problem and our brain can "execute" natural language, i.e. it can answer the question "does this program halt". So if we could build a computer that could do everything our brain can do, it should be able to do this as well. But the thing is our brain can't solve the halting problem with 100% accuracy. Our brain can't even execute regular programs with 100% accuracy. Just remember how often you've stepped through a program in your head and came up with a different result than reality. Our brain is very good at learning, making intuitive connections and applying heuristics, but those things always come with the risk of giving the wrong result.
So could a computer do the same thing? Yes, we can use heuristics and machine learning to approach otherwise unsolvable problems and with that normal programming languages can attempt to solve every problem that can be described in natural language (even the undecidable ones). But just like the brain, those programs will sometimes give wrong results. In fact they will give wrong results much more often as our machine learning algorithms and heuristics aren't nearly as advanced as those of the human brain.