A correspondent raised some interesting questions about thryoid and gut bacteria. She wrote
“Thyroid disease is known to slow down the digestion” I find that I must ask the question, perhaps digestion slowed down FIRST, and the thyroid disease was a result! I don’t know the answer, but know that if we don’t examine both directions — we may miss a significant treatment approach.
Looking at some pubmed articles,
“thyroid hormones increase intestinal motility and transit rate, administration of cholecystokinin increases small bowel motility, diarrhoea occurs in adrenal failure, while mineralocorticoids increase salt and water absorption from the colon.Indeed aldosterone secretion may be increased by the loss of sodium in diarrhoea. The balance between the biochemical regulators is clearly complex, but study of certain endocrine abnormalities has demonstrated their importance in the control of intestinal function.” 
Later studies report “Thyroid diseases may be related to gastrointestinal motility symptoms. Such symptoms can vary in degree and, sometimes, are the only clue of athyroid disease or, at least, the first…. thyroid diseases may be related to symptoms due to digestive motility dysfunction.” 
In terms of demonstrating that the absence or prescence of a single bacteria species in the gut can cause thyroid problems is mentioned in this 2008 article.
“Perhaps the most celebrated example of the difference a single species can make is the ‘mimosine story’ in ruminants. Mimosine is a toxic amino acid found in the leguminous plant, Leucaena leucocephala. Mimosine can cause thyroid problems by being converted to the goitrogen, 3-hydroxy-4(1H)-pyridone, in the rumen. Observations that mimosine-containing plants were toxic to ruminants in some countries but not others led to the discovery of Synergistes jonesii, which metabolises 3-hydroxy-4(1H)-pyridone and protects animals from toxicity.”
In other words, a single species makes a major difference: Synergistes jonesii!
The sweetish article that I found comes from, BMJ in 1923: THE INFLUENCE OF INTESTINAL BACTERIA UPON THE THYROID GLAND, which is well worth a read.
I have talked with people that have thyroid issue run in their family. Since both gut bacteria and DNA are inherited — (which means that the cross product[interaction between DNA and gut bacteria] is also inherited) — this suggests the inheritance could be altered by gut flora alteration (speculation)
If this is the correct approach, then the question arises — how do we alter it? The 1923 articles suggest that lactose is not effective (and this implies that Lactobacillus species may not be effective), thus this leads us to the unusual probiotics. The easiest to obtain is Prescript-Assist which I have referred to earlier in this blog.