This is the type of bitless, sidepull-type halter bridle that I use with my Icelandic Horses / Ponies.
Available from Connie Ouim, Prescott, AZ.
There are several rings; a couple on the side to attach reins, a couple to use as a bit attachment if you wish,
and a ring at the bottom for a lead rope or for reins if you'd like to use it that way.