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

Americans pronounce characterized as KA-ruhk-tuh-rahyzd (/ˈkærəktəˌraɪzd/). Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "The climate of the region is characterized by hot summers".
Record yourself saying "characterized" and play it back. The mic stays on your device — nothing's uploaded.
4 syllables, 11 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
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.
Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

The schwa before R disappears — R becomes the vowel of the syllable. This is the 'er' sound without a distinct vowel before it.

Start with your jaw open wide and your tongue resting low and flat. Glide the front of your tongue up toward the roof of your mouth as your jaw closes halfway.
Same position as S, but add vocal cord vibration. Feel the buzz.

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

The textbook way isn't wrong — it's just not how anyone actually says it.
In "characterized", the "d" 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 first syllable, not the others. Stretch KA — 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.