Touch the tip of your tongue to the roof of your mouth just behind your teeth. Add vocal cord vibration as you release.

Americans pronounce depth as dehpth (/dɛpθ/). You'll hear it in sentences like "The perspective in this drawing gives a sense of depth" or "I criticize the piece for lacking emotional depth or meaning" — more examples below.
Record yourself saying "depth" 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.
Touch the tip of your tongue to the roof of your mouth just behind your teeth. Add vocal cord vibration as you release.

Drop your jaw moderately. Touch the tongue tip behind the bottom front teeth and lift the mid-front part slightly toward the roof.

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

Place the very tip of your tongue slightly between your teeth. Blow air gently around it without voicing.

Click any sentence to see the full breakdown — every link, every reduction, every flap-T.
The textbook way isn't wrong — it's just not how anyone actually says it.
In "depth", the "p" 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.