Tongue pulls back slightly from the D position, blending into R. Sounds close to 'jr'.

Americans pronounce drink as DRIHNGK (/drɪŋk/). In "drink", the "dr" cluster blends into a "jr" sound — a natural American English pronunciation. This is called the DR Sounds Like JR, a small move that separates 'classroom' from 'native'. It comes out as DRIHNGK. You'll hear it in sentences like "Please drink the tea" or "Can I get you another drink?" — more examples below.
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1 syllable, 5 sounds. Explore each sound's mouth shape and how it's made.
Tongue pulls back slightly from the D position, blending into R. Sounds close to 'jr'.

Curl or bunch your tongue without letting the tip touch the roof of your mouth. Brace the sides of your tongue against your upper back teeth, and round your lips slightly.
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.

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The textbook way isn't wrong — it's just not how anyone actually says it.
In "drink", the "dr" cluster blends into a "jr" sound — a natural American English pronunciation. /d/ shifts toward /dʒ/ ("j"), so DR sounds like "jr".
In "drink", 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.