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

Americans pronounce sweater as SWEH-ter (/ˈswɛɾər/). In "sweater", the "t" between vowels sounds like a quick "d" — the tongue briefly taps the ridge behind the upper teeth. This is called the Flap T, and it's one of the defining features of casual American English. It comes out as SWEH·ter. Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "Who took my wool sweater?" or "The wool sweater looked full of hooks" — more examples below.
Record yourself saying "sweater" and play it back. The mic stays on your device — nothing's uploaded.
2 syllables, 5 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

Round your lips into a tight circle. Lift the back of your tongue toward the soft palate and add voice.

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

The textbook way isn't wrong — it's just not how anyone actually says it.
In "sweater", the "t" between vowels sounds like a quick "d" — the tongue briefly taps the ridge behind the upper teeth. /t/ or /d/ becomes a quick tap [ɾ] — sounds like a soft D. The tongue briefly taps the ridge behind the upper teeth.
Stress falls on the first syllable, not the others. Stretch SWEH — keep everything else short and quick.
Americans use a relaxed retroflex R — the tongue curls back rather than rolling. The R is one continuous sound with the vowel before it, not two separate sounds.