This statement, which I think I've seen elsewhere, kinda doesn't make sense to me.

http://www.jcrows.com/iodine.html

Now while the thyroid gland helpfully stores iodine from the blood passing through it every 17 minutes, the gland may also be made to lose that stored iodine if, for example, we take in drinking water to which chlorine is added, or use too much sodium chloride, whose common name is table salt. There is a well-known law of halogen displacement. The halogen group is made up as follows:
                                                         Relative
Halogen                                     Atomic Weight
Fluorine                                              19.
Chlorine                                              35.5
Bromine                                              80.
Iodine                                                127.

The critical activity of any one of these four halogens is in inverse proportion to its atomic weight. This means that any one of the four can displace the element with a higher atomic weight, but cannot displace an element with a lower atomic weight. For example, fluorine can displace chlorine, bromine and iodine because fluorine has a lower atomic weight than the other three. Similarly, chlorine can displace bromine and iodine because they both have a higher atomic weight. Likewise, bromine can displace iodine from the body because iodine has a higher atomic weight. But a reverse order is not possible. A knowledge of this well-known chemical law brings us to a consideration of the addition of chlorine to our drinking water as a purifying agent. We secure a drinking water that is harmful to the body not because of its harmful germ content but because the chlorine content now causes the body to lose the much-needed iodine.
----------------------
Now, Dr. Abraham says:

Iodine increases mobilization of bromine from storage sites with increased urinary excretion of bromide[viii].

Also from others:

Elevated bromide levels were observed in urine and serum samples,[ix] twenty times the levels reported in the literature in normal subjects.[x]

[viii] Abraham, G.E., The historical background of the iodine project. The Original Internist, 12(2):57-66, 2005.
[ix] Sangster, B., Blom, J.L., Sekhuis, V.M., et al, The Influence of Sodium Bromide in Man: A Study in Human Volunteers with Special Emphasis on the Endocrine and the Central Nervous System. Fd. Chem. Toxic., 21:409-419, 1983.
[x] Miller, M.E., Cappon, C.J., Anion-Exchange Chromatographic Determination of Bromide in Serum. Clin. Chem. 30(5):781-783, 1984.
--------------------
So my feeling is that iodine does not get displaced by the lighter halogens.  Iodine just was not around when the thyroid needed it.  Apparently, the thyroid cannot tell the difference between bromine and iodine; so, its only choice was to collect bromine.  Now when iodine is introduced into the body, it is able to chelate the other halogens.

Dr. Kenezy Gyula Korhaz states (http://alobar.livejournal.com/2297226.html) that iodine chelates heavy metals such as mercury, lead, cadmium and aluminum and halogens such as fluoride and bromide, thus decreasing their iodine inhibiting effects[vi] especially of the halogens. Iodine has the highest atomic weight of all the common halogens (126.9). Iodine is the only option when it comes to removing these toxic haloids from the thyroid and even the pineal gland where fluoride concentrates, especially when there is a deficiency in iodine in the body. In an age of increasing radioactivity and toxic poisoning specifically with fluoride[vii], chlorine and bromide, and even mercury, iodine is a necessary mineral to protect us from harm for immediately these toxic substances will increasingly flow out of the body in the urine.
[vi] Sticht, G., Käferstein, H., Bromine. In Handbook on Toxicity of Inorganic Compounds - Seiler HG and Sigel, H Editors, Marcel Dekker Inc, 143-151, 1988.

And this last statement makes more sense to me; than atomic weight displacement.