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

Americans pronounce bird as BURD (/bɜrd/). You'll hear it in sentences like "Her bird has fur" or "The early bird learns the first word" — more examples below.
Record yourself saying "bird" 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.
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.

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.
In "bird", 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.
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.