It's in extremes of feverishness that one has outbursts (4)
I believe the answer is:
fits
'outbursts' is the definition.
('fit' can be a synonym of 'outburst')
'it's in extremes of feverishness that one' is the wordplay.
'it's in' is an insertion indicator.
'extremes of' means to remove the middle letters (only the letters at the extremes of the word).
'that one' becomes 'it' (I can't explain this - if you can you should give a lot more credence to this answer).
'feverishness' with its middle removed is 'fs'.
'fs' placed around 'it' is 'FITS'.
'has' is the link.
Can you help me to learn more?
(Other definitions for fits that I've seen before include "Seizures" , "Is of the right shape and size" , "Sudden uncontrollable attacks" , "short bouts" , "crises once" .)