Drop your jaw moderately. Touch the tongue tip behind the bottom front teeth and lift the mid-front part slightly toward the roof.

Americans pronounce exhale as ehks-HAYL (/ɛksˈheɪl/). The L in "exhale" is a dark L — the back of the tongue rises toward the soft palate, adding a small "uh" quality before the L. This is called the Dark L vs Light L, and it's one of the defining features of casual American English. It comes out as ehks·HAYL. Stress falls on the second syllable — keep everything else short and quick. You'll hear it in sentences like "Inhale the healthy air and exhale the hate".
Record yourself saying "exhale" 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.
Drop your jaw moderately. Touch the tongue tip behind the bottom front teeth and lift the mid-front part slightly toward the roof.

Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

Push a stream of air from your throat through your open mouth. No tongue or lip contact.

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.
Keep the tongue tip down and pull the back of the tongue up toward the throat. The 'dark' sound comes from the back.

The textbook way isn't wrong — it's just not how anyone actually says it.
The L in "exhale" is a dark L — the back of the tongue rises toward the soft palate, adding a small "uh" quality before the L. Dark L adds a small schwa-like "uh" before the L. The back of the tongue lifts toward the soft palate.
Stress falls on the second syllable, not the others. Stretch HAYL — keep everything else short and quick.