Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

Americans pronounce curve as KURV (/kɜrv/). You'll hear it in sentences like "Cover the curve with a clever move" or "Turn the curve and observe the world" — more examples below.
Record yourself saying "curve" 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.
Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

Flare your lips and push them away from the face. Lift the middle of your tongue toward the roof of the mouth.

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.

The textbook way isn't wrong — it's just not how anyone actually says it.
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.