Press your lips together, add vocal cord vibration, then release.

Americans pronounce birdwatching as BURD-wah-chuhng (/ˈbɜrdˌwɑtʃɪŋ/). Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "He enjoys birdwatching and keeping a log of the species he sees".
Record yourself saying "birdwatching" and play it back. The mic stays on your device — nothing's uploaded.
3 syllables, 8 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Press your lips together, 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.

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

Round your lips into a tight circle. Lift the back of your tongue toward the soft palate and add voice.

Relax your lips and drop your jaw significantly. The tongue tip lightly touches behind the bottom front teeth and the back part of the tongue presses down a little to create more dark space in the back of the mouth.

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

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
Lift the back of your tongue to the soft palate. Lower your soft palate to let air flow through your nose.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "birdwatching", the "d" is not released — the articulators get into position but hold without the burst of air. Air stops but there's no release burst — the articulators hold position.
Stress falls on the first syllable, not the others. Stretch BURD — keep everything else short and quick.
Don't pronounce the second syllable too fully. The unstressed syllable reduces to a schwa — the lazy "uh" sound — in casual speech.
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.