Lift your bottom lip to touch the very bottom of your top front teeth. Blow air through this contact point without voicing.

Americans pronounce fact as FAKT (/fækt/). You'll hear it in sentences like "It's a fact that we're behind schedule" or "It's a fact that you can't really ignore" — more examples below.
Record yourself saying "fact" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 4 sounds. Explore each sound's mouth shape and how it's made.
Lift your bottom lip to touch the very bottom of your top front teeth. Blow air through this contact point without voicing.

Drop the jaw noticeably. Keep the body of the tongue low and forward, and don't let the back of the tongue raise toward the soft palate. Pull the lip corners back slightly, almost a starting smile.

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

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 "fact", 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.