Pull the corners of your lips back slightly. Arch the middle-front of your tongue high toward the roof of the mouth.

Americans pronounce eagle as EE-guhl (/ˈigəl/). The L in "eagle" 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 why Americans sound more relaxed than the textbook. It comes out as EE·guhl. Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "The eagle soared high above the mountains looking for prey".
Record yourself saying "eagle" and play it back. The mic stays on your device — nothing's uploaded.
2 syllables, 4 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Raise the back of your tongue to touch the soft palate. Add vocal cord vibration, then release.

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
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 "eagle" 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 first syllable, not the others. Stretch EE — 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.