Touch the front of your tongue to the roof of your mouth, then release into a 'sh' position. Flare your lips.

Americans pronounce chart as CHART (/tʃɑrt/). You'll hear it in sentences like "Check the chart" or "The child chased the chalk across the chart" — more examples below.
Record yourself saying "chart" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 3 sounds. Explore each sound's mouth shape and how it's made.
Touch the front of your tongue to the roof of your mouth, then release into a 'sh' position. Flare your lips.

Open wide for the 'ah' vowel. Lift the tongue back and up while flaring the lips for the 'r'.
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 "chart", 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.
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.