Drop your jaw slightly with relaxed lips. Touch the tongue tip behind the bottom front teeth and arch the top-front toward the roof.

Americans pronounce exhibit as ihg-ZIH-buht (/ɪgˈzɪbət/). Stress falls on the second syllable — keep everything else short and quick. You'll hear it in sentences like "The exhibit was marked and entered into evidence" or "I wish to visit the exhibit in the middle of spring" — more examples below.
Record yourself saying "exhibit" and play it back. The mic stays on your device — nothing's uploaded.
3 syllables, 7 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.

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
Touch the tip or front edge of your tongue to the roof of your mouth just behind your teeth. Keep your jaw relaxed. Stop the air, then release with a puff.

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 "exhibit", the "t" 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 second syllable, not the others. Stretch ZIH — 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.