The most common and most likely answer to this crossword clue is the 5 letter word STORK
I've seen this crossword clue in The Guardian
S
T
O
R
K
More clues leading to the result STORK
Buddy explains!

The straight part
We know that "Baby bird" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case baby bird = stork.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 STORK mean?
STORK noun- A large wading bird with long legs and a long beak of the family Ciconiidae.
- (children's folklore) The mythical bringer of babies to families, or good news.
- The seventeenth Lenormand card.