Daughter browsed around district, almost made to seem small (7)
I believe the answer is:
dwarfed
'made to seem small' is the definition.
(to dwarf something is to make it seem smaller)
'daughter browsed around district almost' is the wordplay.
'daughter' becomes 'd' (genealogical abbreviation).
'browsed' becomes 'fed' (browse can mean to feed on vegetation).
'around' means one lot of letters goes inside another.
'district' becomes 'ward' (council district).
'almost' means to remove the last letter (most of the word but not all of it).
'd'+'fed'='dfed'
'ward' with its last letter taken away is 'war'.
'dfed' placed around 'war' is 'DWARFED'.
(Another definition for dwarfed that I've seen is " Made to look small".)