Exaggerate European doctor touching designer up (9)
I believe the answer is:
embroider
'exaggerate' is the definition.
(embroidering is a kind of exaggerating)
'european doctor touching designer up' is the wordplay.
'european' becomes 'e' (abbreviation e.g. EU).
'doctor' becomes 'mb' (Bachelor of Medicine, Medicinae Baccalaureus**).
'touching' becomes 're' (I've seen this in other clues).
'designer' becomes 'dior' (dior is a designer brand).
'up' is a reversal indicator (in down clue: letters go upwards).
're'+'dior'='redior'
'redior' back-to-front is 'roider'.
'e'+'mb'+'roider'='EMBROIDER'
(Other definitions for embroider that I've seen before include "Embellish (with needlework?)" , "Elaborately sew" , "Do decorative needlework" , "Ornament with needlework" , "stitch" .)