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

Americans pronounce earth's as URTHS (/ˈɜrθs/). You'll hear it in sentences like "The greenhouse effect traps heat in the earth's atmosphere" or "Geologists study the composition and structure of the earth's crust" — more examples below.
Record yourself saying "earth's" 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 push them away from the face. Lift the middle of your tongue toward the roof of the mouth.

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

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

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.
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.