I'm not familiar with a 'headpiece' but I think you mean the bridle. This is used to control the horse's head and keeps the bit in the horse's mouth.
A "bridle path".Makes a channel in a thick mane for the headpiece of the bridle to sit in.
It is called a bridal patch or bridal path for the headpiece of the bridal to sit when it is on.
A gele is a traditional headpiece worn by an African bride.
Cleopatra's headpiece or crown was called a "uraeus".
This question is a fail
Figurehead
it is a headpiece worn by a bishop.
A part of the harness which goes over the horse's neck.
A head piece
I suggest looking at Amazon
Fascinator
Laurel wreath/crown. Both are correct. :)