Native American swiftly catches horse's head (6)
I believe the answer is:
apache
'native american' is the definition.
(group of Native American tribes)
'swiftly catches horse's head' is the wordplay.
'swiftly' becomes 'apace' (synonyms).
'catches' is an insertion indicator.
'head' suggests taking the first letters.
The first letter of 'horses' is 'h'.
'apace' enclosing 'h' is 'APACHE'.
(Other definitions for apache that I've seen before include "A native N. American people" , "Parisian ruffian - Indian" , "language" , "ruffian in capital of France" , "Geronimo's people" .)