Place your tongue tip between or behind your front teeth, turn your vocal cords on, and push air through the gap.
How to pronounce that's in American English
Americans pronounce that's as dhats (/ðæts/). In fast speech, the vowel in "that's" vanishes — the "ih" is completely elided, leaving only a quick "ts" cluster — this is a feature of casual, connected speech; in careful speech, the vowel is retained. This is called the Short Contractions (it's, that's), a quick, quiet beat that keeps content words in focus. It comes out as dhats. You'll hear it in sentences like "That's a bad place to put your bed" or "That's a real lifesaver, thank you" — more examples below.
Now you try.
Record yourself saying "that's" and play it back. The mic stays on your device — nothing's uploaded.
Every sound in "that's".
1 syllable, 4 sounds. Explore each sound's mouth shape and how it's made.
Drop the jaw noticeably. Keep the body of the tongue low and forward, and don't let the back of the tongue raise toward the soft palate. Pull the lip corners back slightly, almost a starting smile.

Touch the tip or front edge of your tongue to the roof of your mouth just behind your teeth. Keep your jaw relaxed. Stop the air, then release with a puff.

Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

Hear "that's" in the wild.
Click any sentence to see the full breakdown — every link, every reduction, every flap-T.