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

Americans pronounce thirds as THURDZ (/θɜrdz/). You'll hear it in sentences like "The constitutional amendment requires a two-thirds majority to pass".
Record yourself saying "thirds" 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 the very tip of your tongue slightly between your teeth. Blow air gently around it without voicing.

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

Same position as S, but add vocal cord vibration. Feel the buzz.

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.