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

Americans pronounce drain as DRAYN (/dreɪn/). In "drain", the "dr" cluster blends into a "jr" sound — a natural American English pronunciation. This is called the DR Sounds Like JR, and it's one of the defining features of casual American English. It comes out as DRAYN. You'll hear it in sentences like "You must pull hard on the pole to drain the pool" or "We hired a plumber to fix the clogged drain in the bathroom" — more examples below.
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1 syllable, 4 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.
Start with your jaw slightly open and the front of your tongue forward and slightly up. Glide upward, your jaw closes a little more and your tongue arches higher toward the roof of the mouth.
Touch the tip or front edge of your tongue to the roof of your mouth behind your teeth. Air flows through your nose.

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