Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

Americans pronounce suite as SWEET (/swit/). You'll hear it in sentences like "The hotel suite smells like sweet fruit" or "Staying in this suite is a very sweet deal" — more examples below.
Record yourself saying "suite" 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.
Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

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

Pull the corners of your lips back slightly. Arch the middle-front of your tongue high toward the roof of the mouth.

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.

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 "suite", 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.