Flare your lips and lift the mid-front tongue close to the roof of your mouth. Blow air through without voicing.

Americans pronounce shout as SHOWT (/ʃaʊt/). You'll hear it in sentences like "Do not shout loud" or "The crowd found the sound of the loud shout profound" — more examples below.
Record yourself saying "shout" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 3 sounds. Explore each sound's mouth shape and how it's made.
Flare your lips and lift the mid-front tongue close to the roof of your mouth. Blow air through without voicing.

Start with a dropped jaw and flat tongue. Glide into a relaxed, slightly rounded lip position as the back of the tongue stretches up.
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 "shout", 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.