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?

 A: 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:
Damien Woods, Turlough Neary, "On the time complexity of 2-tag systems and small universal Turing machines" (2006) 
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. 
A: Because we can easily invent as many small variations of Turing-complete models of computation as we like (see comments below the question), an answer to this question should try to concentrate on relevant (and Turing-complete) models, i.e. models that have either been investigated in illuminating non-trivial ways, or are important for better understanding of actually available computing resources.
I have been exposed in non-trivial ways to tape based Turing machines, register machines and pointer machines. It seem like the wikipedia article on abstract machines is intended to give an overview for related Turing machine equivalent models, but in its current form it is mainly a collection of useful keywords and links.

I'm currently looking for models and investigations related to machines limited to write once read many (WORM) memory for large amounts of data. None of the abstract machine models I found so far investigated these. Is it possible to create a model of such a machine that is equivalent to a Turing machine in the sense of the question above? (Edit: It looks like it was proved recently that Wang B-machines achieve this. I haven't read the paper yet.) This question seems to be both non-trivial and interesting to me, contrary to the comments below the question, which is the main reason why I wrote this answer.
