How far is the Thirty Meter Telescope estimated to see?

All the way.

Seriously, just about any decent-sized telescope can see galaxies that are billions of light-years away. Seeing a distance is not the problem, it is getting fine details of things that are ten billion light years away. The TMT will have 0.007" resolution, about seven times better than Hubble. And it can get that resolution 144 times faster than Hubble, so it can look at 144 times more stuff in the same time.

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(later edit)
I don't know what you mean by "resolution of 4K". The Wide Field Optical Spectrometer will be equivalent to about 904 pixels by 904 pixels. It's really designed for pin-point observations of single objects.

The TMT will be able to see as far as the "edge of the universe"; really that means the furthest galaxies in the observable universe. There's just nothing else farther away that gives off light. Call it 13.3 billion light years.

Current large telescopes can "see" (as fuzzy little patches) the very earliest galaxies.

The new giant telescopes will not only be able to see those fuzzy patches, but will also be able to take spectra.

Those are good answers = about 13.6 billion light years.

To the edge of our visible universe.