Relax your lips and drop your jaw significantly. The tongue tip lightly touches behind the bottom front teeth and the back part of the tongue presses down a little to create more dark space in the back of the mouth.

Americans pronounce alternate as AHL-ter-nuht (/ˈɔltərnət/). The L in "alternate" 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, the kind of sound shift that makes everyday speech feel effortless. It comes out as AHL·ter·nuht. Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "The highway traffic was terrible, so I took an alternate route".
Record yourself saying "alternate" 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.
Relax your lips and drop your jaw significantly. The tongue tip lightly touches behind the bottom front teeth and the back part of the tongue presses down a little to create more dark space in the back 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.

Touch the tip or front edge of your tongue to the roof of your mouth behind your teeth. Air flows through your nose.

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.

The textbook way isn't wrong — it's just not how anyone actually says it.
The L in "alternate" 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.
In "alternate", 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 first syllable, not the others. Stretch AHL — 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.