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

Americans pronounce contagious as kuhn-TAY-juhs (/kənˈteɪdʒəs/). Stress falls on the second syllable — keep everything else short and quick. You'll hear it in sentences like "His enthusiasm is contagious".
Record yourself saying "contagious" and play it back. The mic stays on your device — nothing's uploaded.
3 syllables, 8 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
Touch the tip or front edge of your tongue to the roof of your mouth behind your teeth. Air flows through your nose.

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.

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 front of your tongue to the roof of your mouth, then release into a 'zh' position. Add vocal cord vibration.

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

The textbook way isn't wrong — it's just not how anyone actually says it.
Stress falls on the second syllable, not the others. Stretch TAY — 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.