At the bottom of the page are links to some articles on the internet about "accepting the bit".
One of the articles has the following requirements for acceptance of the bit:
mentally and physically relaxed
going forward from the rider's aids
going straight on a straight line or bending correctly on a circle
in a rhythm and therefore in balance
in a natural outline, flexing at the poll with no resistance in his mouth
Some horses, of any breed, and of Icelandic Horses, are started too fast, too soon, and it may be a little harder to work with a horse who has already been started in this manner. But it's not impossible to turn one of these horses around. It will take time and dedication to the commitment, and placing the horse first and foremost in the equation.
Be careful of the term "sensitive" when applied to a horse that has been trained in this manner. It may mean that the horse doesn't know how to use the bit as a means of communication, may bolt or run. In any case, it is an incorrectly or under-educated horse.
It may be good to start with Passenger Lessons (Parelli Natural Horsemanship, PNH) of which some information can be found on the internet and also in the IceHorses email list archives (Icelandic Horse Discussion List); search under: Passenger Lessons and / or Self-Carriage.
It may also be a consideration to re-start the horse bitless, either with a sidepull, rope halter, or bitless-type bridle.
The horse will learn not to lean on the bit, to be responsible for it's own carriage, and to be "light".
It truly is an incredible feeling to change one of the heavy horses over to a light and responsive horse. Icelandic Horses love to learn to carry themselves, and not be dependant on so much contact on the bit.