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

Americans pronounce quit as KWIHT (/kwɪt/). You'll hear it in sentences like "I didn't think he would actually quit".
Record yourself saying "quit" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 4 sounds. Explore each sound's mouth shape and how it's made.
Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

Round your lips into a tight circle. Lift the back of your tongue toward the soft palate and add voice.

Drop your jaw slightly with relaxed lips. Touch the tongue tip behind the bottom front teeth and arch the top-front toward the roof.

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.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "quit", the "t" is not released — the articulators get into position but hold without the burst of air. Air stops but there's no release burst — the articulators hold position.