Touch the tip of your tongue to the roof of your mouth just behind your teeth. Add vocal cord vibration as you release.

Americans pronounce distinguish as duh-STIHNGG-wihsh (/dəˈstɪŋgwɪʃ/). Stress falls on the second syllable — keep everything else short and quick. You'll hear it in sentences like "I find it difficult to distinguish between similar sounding words".
Record yourself saying "distinguish" and play it back. The mic stays on your device — nothing's uploaded.
3 syllables, 10 sounds. Tap a syllable to jump to its row, then 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.

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.

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. Add vocal cord vibration, then release.

Round your lips into a tight circle. Lift the back of your tongue toward the soft palate and add voice.

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

Flare your lips and lift the mid-front tongue close to the roof of your mouth. Blow air through without voicing.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "distinguish", the "g" 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.
Stress falls on the second syllable, not the others. Stretch STIHNGG — keep everything else short and quick.
Don't pronounce the first syllable too fully. The unstressed syllable reduces to a schwa — the lazy "uh" sound — in casual speech.