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

Americans pronounce sink as SIHNGK (/sɪŋk/). You'll hear it in sentences like "Does he think the soap is in the sink?" or "I think the dishes are still in the sink" — more examples below.
Record yourself saying "sink" 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.
Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

Drop your jaw slightly with relaxed lips. Touch the tongue tip behind the bottom front teeth and arch the top-front toward the roof.

Lift the back of your tongue to the soft palate. Lower your soft palate to let air flow through your nose.

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

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 "sink", the "k" 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.