Let's first take note that no actual Turing-complete machine has ever been built, nor can one ever be built. Modern computers are good enough approximations for most purposes, but they are in reality just very large finite state machines. A true Turing machine requires infinite storage space.
The infinite storage space of the Turing machine model is mirrored in the (computationally equivalent) lambda calculus by the capacity to define an infinite number of arbitrarily-named variables. This suggests a natural connection to human languages: the potentially-infinite set of variables in lambda calculus corresponds to the set of words in a human language, while human grammars correspond to the reduction rules of lambda calculus. Transferring that back to the Turing machine model, the infinite storage tape of a Turing machine corresponds to the potential vocabulary of a human language, while the state transition rules correspond to a grammar.
So, the first thing you need to make a human language "Turing complete"--capable of referring to or describing anything that can be referred to or described--is an infinite vocabulary! That's clearly impossible, but like real computers approximate Turing machines, we can approximate a "Turing complete conlang" simply by ensuring there is some way to create new lexical items--either strictly by as-needed de-novo coinage, or oligosynthesis, or whatever other word-formation techniques you like--and then abstract away from the vocabulary and looking at the rules for how that vocabulary is used.
Continuing the analogy, the fact that there are two such radically different formulations for computation that are nevertheless provably identical in expressive power (the lambda calculus and Turing machine--and, in fact, even more different formalisms than that, like the SKI combinator calculus, NAND machines and the One Instruction Set Computer, etc.) suggests that there may not be any single simplest set of grammar constructs that make an abstract language "complete".
Empirically, however, if you put stock in David Gil's work on so-called IMA (Isolating-Monocategorial-Associative) language, the only necessary and sufficient requirement for effective human communication (or as he puts it, enough grammar "to sail a boat") is the existence of a generic "association operator"--a way to say "the things represented by these two sub-phrases are related somehow"--and literally everything else can be handled by pragmatics.
So, you need the ability to introduce whatever words you find that you need in a given situation, and you need at least one operator that can specify arbitrary relations between things (e.g., the generic English preposition "of")--which could have a surface expression as simple as juxtaposition. Everything else is sugar, that just makes figuring out the pragmatic details easier.
