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

Americans pronounce birthdays as BURTH-dayz (/ˈbɜrθˌdeɪz/). Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "She enjoys baking and decorating elaborate cakes for birthdays".
Record yourself saying "birthdays" and play it back. The mic stays on your device — nothing's uploaded.
2 syllables, 6 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Touch the tip of your tongue to the roof of your mouth just behind your teeth. Add vocal cord vibration as you release.

Start with your jaw slightly open and the front of your tongue forward and slightly up. Glide upward, your jaw closes a little more and your tongue arches higher toward the roof of the mouth.
Same position as S, but add vocal cord vibration. Feel the buzz.

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 BURTH — 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.