Push a stream of air from your throat through your open mouth. No tongue or lip contact.

Americans pronounce hurt as HURT (/hɜrt/). You'll hear it in sentences like "Certainly, the service was worthy of hurt" or "Beck hurt his back while riding a heavy bike" — more examples below.
Record yourself saying "hurt" 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.
Push a stream of air from your throat through your open mouth. No tongue or lip contact.

Flare your lips and push them away from the face. Lift the middle of your tongue 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 "hurt", 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.
Americans use a relaxed retroflex R — the tongue curls back rather than rolling. The R is one continuous sound with the vowel before it, not two separate sounds.