From what little I understand the specialness of a Turing machine is that it can be used as a universal calculator, not that it can be used as a universal calculator fast.This depends on how you want to understand fast. A Turing machine updates only a few bits per cycle while a modern computer easily updates many thousands if not millions of bits per cycle. Also a Turing machine access its memory very slowly because the head moves one cell per cycle, good luck reading a bit stored five gigabytes away from the current position of the head. So yes, in that sense a Turing machine is slow, especially random access memory is big advantage in practice, but in theoretic terms having to scan through the entire memory just gets swept under the rug of polynomial factors.
On the other hand you can build a Turing machine that emulates your modern computer even though it might need a trillion cycles to emulate just a single cycle of it. But mathematically you can just factor this out, you call a trillion cycles just one cycle and now both are the same speed, mathematically of course, not any physical implementation.
The other way that you allude to, using many Turing machines in parallel, is probably more realistic. You can probably build a small Turing machine with the equivalent of a few hundred or maybe few thousand gates. And you can program them to simulate a single logic gate in maybe ten or so cycles. Now rebuild you real computer with all gates replaced by those small Turing machines.
My gut feeling is that you could probably build a 80s or 90s era processor using current technology. A 1982 Intel 80286 has 134,000 transistors, that puts the number of gates well below 100,000 while a GeForce RTX 4090 has 16,384 shader cores which are way more powerful than our tiny Turing machine to simulate a single logic gate needs to be. So if you are willing to give up 30 years of Moore's law, you can probably have a Turing machine powered computer.
Heck [2], we have transistor level simulations of old processors [1], somebody with enough enthusiasm could probably do this at the gate level, then replacing gates with Turing machine simulations would be easy, and finally you would have to get them running on a GPU for decent speed, but I am not sure if GPUs would be any good for that kind of task, I have never programmed one. But this would be a processor running some code, simulated at the gate level with the gates implemented by Turing machines simulated on a real processor.
[1] http://visual6502.org/JSSim/index.html
[2] Is there any good replacement for heck, I think it has a negative connotation? But maybe stronger than look.