Tongue pulls back slightly from the T position, blending into R. Sounds close to 'chr'.

Americans pronounce transplant as TRAN-splant (/ˈtrænsˌplænt/). In "transplant", the "tr" cluster blends into a "chr" sound — a natural American English pronunciation. This is called the TR Sounds Like CHR, a hallmark of natural-sounding American speech. It comes out as TRAN·SPLANT. Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "The surgeon performed a complex heart transplant successfully".
Record yourself saying "transplant" and play it back. The mic stays on your device — nothing's uploaded.
2 syllables, 10 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Tongue pulls back slightly from the T position, blending into R. Sounds close to 'chr'.

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.
The tongue relaxes down in the back and the corners of the lips relax before the consonant. This adds a schwa-like 'uh' relaxation after the /æ/. Think of it as 'relaxing out of the vowel' — it is no longer a pure /æ/ sound.

Touch the tip or front edge of your tongue to the roof of your mouth behind your teeth. Air flows through your nose.

Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

Press your lips together to stop the air, then release. No vocal cord vibration.

Place the tip of your tongue against the alveolar ridge just behind your top front teeth, the same contact point as /t/, /d/, and /n/. The difference is what happens to the air: for /l/, you let it flow continuously around the <em>sides</em> of the tongue (that's why /l/ is called a lateral). Turn your voice on the whole time. Lips stay relaxed, no rounding or flaring. For the Dark L variant at the end of a syllable, also pull the back of the tongue up and back toward the soft palate.

The tongue relaxes down in the back and the corners of the lips relax before the consonant. This adds a schwa-like 'uh' relaxation after the /æ/. Think of it as 'relaxing out of the vowel' — it is no longer a pure /æ/ sound.

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.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "transplant", the "tr" cluster blends into a "chr" sound — a natural American English pronunciation. /t/ shifts toward /tʃ/ ("ch"), so TR sounds like "chr".
In "transplant", the "a" vowel before M or N raises and fronts toward [eə] — the tongue pulls up and forward, breaking the vowel into a tense glide as it anticipates the nasal. The "/æ/" vowel raises and fronts before M or N — tongue pulls up and forward, producing a tense [eə] glide (between /e/ and /ə/). Not a pure /æ/.
Stress falls on the first syllable, not the others. Stretch TRAN — keep everything else short and quick.