Cryptic crossword clue
Overly optimistic girl gets infected, but antibiotics initially not needed 4-6

The most common and most likely answer to this crossword clue is the 11 letter word ROSE-TINTED

I've seen this crossword clue in Independent
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Buddy explains!

The straight part
We know that "Overly optimistic" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case overly optimistic = rose-tinted.
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.

Crossword Answer definitions

ROSE-TINTED verb
  1. To look through rose-tinted glasses at; to view or describe as better than it actually is or was.
ROSE-TINTED verb
  1. To tint with the color rose; to suffuse with a rosy hue.

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