Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

Americans pronounce karate as kuh-RAH-tee (/kəˈrɑɾi/). In "karate", the "t" between vowels sounds like a quick "d" — the tongue briefly taps the ridge behind the upper teeth. This is called the Flap T, the kind of sound shift that makes everyday speech feel effortless. It comes out as kuh·RAH·tee. Stress falls on the second syllable — keep everything else short and quick. You'll hear it in sentences like "He is a black belt in karate and teaches self-defense".
Record yourself saying "karate" and play it back. The mic stays on your device — nothing's uploaded.
3 syllables, 6 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Curl or bunch your tongue without letting the tip touch the roof of your mouth. Brace the sides of your tongue against your upper back teeth, and round your lips slightly.
Relax your lips and drop your jaw significantly. The tongue tip lightly touches behind the bottom front teeth and the back part of the tongue presses down a little to create more dark space in the back of the mouth.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "karate", the "t" between vowels sounds like a quick "d" — the tongue briefly taps the ridge behind the upper teeth. /t/ or /d/ becomes a quick tap [ɾ] — sounds like a soft D. The tongue briefly taps the ridge behind the upper teeth.
Stress falls on the second syllable, not the others. Stretch RAH — keep everything else short and quick.
Don't pronounce the first syllable too fully. The unstressed syllable reduces to a schwa — the lazy "uh" sound — in casual speech.