TITLE: What Turing-Complete models of computation carry a notion of time complexity that "agrees" with that of Turing Machines?
QUESTION [10 upvotes]: 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?

REPLY [6 votes]: 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.