Press your lips together. Air flows through your nose. Vocal cords vibrate.

Americans pronounce mesmerizing as MEHZ-muh-rahy-zuhng (/ˈmɛzməˌraɪzɪŋ/). Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "The immense mountain was mesmerizing to me".
Record yourself saying "mesmerizing" and play it back. The mic stays on your device — nothing's uploaded.
4 syllables, 10 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
The schwa before R disappears — R becomes the vowel of the syllable. This is the 'er' sound without a distinct vowel before it.

Start with your jaw open wide and your tongue resting low and flat. Glide the front of your tongue up toward the roof of your mouth as your jaw closes halfway.
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 MEHZ — keep everything else short and quick.
Don't pronounce the first syllable too fully. The unstressed syllable reduces to a schwa — the lazy "uh" sound — in casual speech.