I don't have reputation enough to comment, but I don't think it's a stupid question at all (though it may not be appropriate for cstheory). The answer is no, it is not at all guaranteed; in fact, there are proofs to the contrary. For single-tape Turing machines, there apparently is a result on the complexity of optimal sorting routines by Juraj Wiedermann (1992): Optimal Algorithms for Sorting on Single-tape Turing machines. Unfortunately, I'm not able to access a copy. (Note that there's an answer on StackOverflow, asserting that the lower bound on big-O complexity for a single-tape machine has been proved to be $O(n^2 log(n))$, but I haven't been able to verify that.)
More recent work is available which discusses off-line Turing machines -- these are Turing machines which, besides the usual working tape, have access to an "input tape" which contains the input to the problem. The input tape is read-only, and the head cannot be moved beyond the bounds of the input. For these machines, complexity of an optimal sorting routine is given by Petersen (2008): Element Distinctness and Sorting on One-Tape Off-Line Turing Machines.