What Turing-Complete models of computation carry a notion of time complexity that “agrees” with that of Turing Machines?

Certain models of computation are technically Turing-Complete, but cannot feasibly simulate a Turing Machine within the usual time constraints we hope for. One example of this is Godel's recursive functions: the computable function $f(x) = 2x$ is implemented by calling the successor function $2x$ times, which intuitively takes $O(2^x)$ time.

So, my question:

What Turing-complete models of computation, or simplistic programming languages (Turing tar pits), can compute every computable function with only a worst-case polynomial-time blowup in time complexity over the fastest Turing machine that computes that same function?

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Aren't there hundreds of examples? This is the robustness of $P$ as a complexity class, that we can often (and indeed usually) simulate a different model of computability in our own, with at most polynomial time cost. –  Joel David Hamkins Sep 3 '13 at 14:41
I imagine there are hundreds of examples of languages without this problem (every modern programming language, for instance) but I don't know of any simplified, theory-driven model of computation that doesn't have this problem. Recursive functions, Lambda calculus, tag systems, and brainfuck all have computable functions that they implement in exponential time, but a Turing machine could implement these functions in polynomial time. I'm looking for an example along the lines of these simplified models of computation. –  GMB Sep 3 '13 at 19:23
Well, some Turing machine models are also weak in this way. For example, if you have only two symbols and a single tape, and are forced to use unary notation (so that you know when the input ends), then it will take exponential time to simulate the usual machines, which are more powerful. –  Joel David Hamkins Sep 3 '13 at 19:48
Okay, good point - I should've specified that I'm only "counting" the Turing machine models that have sufficient alphabet size to solve all problems as quickly as possible. –  GMB Sep 3 '13 at 19:56
Meanwhile, I would point out that even the ordinary notions of Turing machine are idealized concepts, and not about what we can actually build in the real world. We cannot expect to build a Turing machine in the physical world that would undertake computation of appreciable size, since once the paper tape was large enough, it would be subject to gravitational forces. Is it in orbit? If coiled, once the mass became a certain size it would collapse into a black hole. If uncoiled, it would likely tear. Long story short: Turing computability is a notion of idealized computation. –  Joel David Hamkins Sep 4 '13 at 12:06

One of the simplest model that has recently been proved to be an efficient simulator (polynomial time slowdown) of Turing machines are 2-tag systems:

Abstract: We show that 2-tag systems efficiently simulate Turing machines. As a corollary we find that the small universal Turing machines of Rogozhin, Minsky and others simulate Turing machines in polynomial time. This is an exponential improvement on the previously known simulation time overhead and improves a forty year old result in the area of small universal Turing machines.

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