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

Americans pronounce cowgirl as KOW-gurl (/ˈkaʊˌgɜrl/). Stress falls on the first syllable — keep everything else short and quick.
Record yourself saying "cowgirl" and play it back. The mic stays on your device — nothing's uploaded.
2 syllables, 5 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Raise the back of your tongue to touch the soft palate. Add vocal cord vibration, then release.

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

Keep the tongue tip down and pull the back of the tongue up toward the throat. The 'dark' sound comes from the back.

The textbook way isn't wrong — it's just not how anyone actually says it.
Stress falls on the first syllable, not the others. Stretch KOW — keep everything else short and quick.
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.