The most common and most likely answer to this crossword clue is the 8 letter word LEUKEMIA
I've seen this crossword clue in The Guardian
L
E
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K
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M
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A
Buddy explains!

The straight part
We know that "US-style disease" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case us-style disease = leukemia.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does LEUKEMIA mean?
LEUKEMIA noun- A type of malignancy affecting the blood cells or blood-forming tissues.
- Any specific form or type of cancer of the blood-forming tissues.