Touch the front of your tongue to the roof of your mouth, then release into a 'sh' position. Flare your lips.

Americans pronounce church as CHURCH (/tʃɜrtʃ/). It's 3 sounds in 1 syllable. The R curls back and blends into the vowel — they sound as one continuous unit.
Record yourself saying "church" 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.
Touch the front of your tongue to the roof of your mouth, then release into a 'sh' position. Flare your lips.

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

Touch the front of your tongue to the roof of your mouth, then release into a 'sh' position. Flare your lips.

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.