Caterpillar turning an age to undergo rapid development (7)
I believe the answer is:
burgeon
'undergo rapid development' is the definition.
'caterpillar turning an age' is the wordplay.
'caterpillar' becomes 'grub' (I've seen this before).
'turning' says the letters should be written in reverse.
'an age' becomes 'eon' (eon is a kind of age).
'grub' written backwards gives 'burg'.
'burg'+'eon'='BURGEON'
'to' acts as a link.
(Other definitions for burgeon that I've seen before include "swell" , "sprout" , "produce sprouts" , "develop quickly" , "Produce buds - flourish" .)