Press your lips together to stop the air, then release. No vocal cord vibration.

Americans pronounce private as PRAHY-vuht (/ˈpraɪvət/). Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "He lives on a quiet, private street".
Record yourself saying "private" and play it back. The mic stays on your device — nothing's uploaded.
2 syllables, 6 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Press your lips together to stop the air, then release. No vocal cord vibration.

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.
Start with your jaw open wide and your tongue resting low and flat. Glide the front of your tongue up toward the roof of your mouth as your jaw closes halfway.
Lift your bottom lip so its inner edge (where the wet part meets the dry part) touches the very bottom of your top front teeth. Add vocal cord vibration as you blow air through.

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
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 "private", 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.
Stress falls on the first syllable, not the others. Stretch PRAHY — 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.