There's nothing wrong with the veggies you mention. The only reason they might cause bloating is because they are chock full of fiber and if you aren't used to eating a lot fiber, it can cause you to bloat.
Broccoli is a goitrogen, which can interfere with thyroid function, but only when eaten raw. Cooking foods that are considered goitrogens, takes away the goitrogenic properties.
You mention gluten in your title, then rye vs white bread. Actually, real rye bread would be healthier than white, if it's made from whole grain. There's no need to eliminate gluten from your diet unless you are allergic or sensitive to it.
You know I can't eat white bread or pasta because it makes me feel so bloated, so I just don't eat it.
I am also lactose intolerant so i just stay away from milk & ice cream. I can eat veggies, but I only like potatoes, corn, & green beans.
If I get a double meat burger , it is without the buns. I rarely eat bread.
I know what bothers my stomach, but as far as thyroid I was never told what to eat and what not to eat. If it bothers my stomach I won't eat it. I can't tell if what I do eat affects my thyroid levels or not. I love grilled chicken or BBQ chicken & wings.....I also eat a lot of tun & soup.
I had a RAI & then a TT 18yrs ago.
They don't tell you what you should and shouldn't be eating because there are no proven foods you should stay away from. There are theories out there about what should be avoided, but they are just that...theories.
Have you been diagnosed with Hashi's?
The veggies you mention I eat all the time, and they give me a lot less bloating than many other foods. My advice is that if something causes YOU a problem personally, avoid it, by all means. However, if you avoid foods because someone tells you it will improve your thyroid condition or because it has an adverse effect on someone else, you are probably going to be very disappointed.
Many of us have not changed out diets or lifestyles on iota to accomodate our thyroid conditions.
If you're struggling with weight issues, it's very possible that you are undermedicated. It can be almost impossible to lose weight if thyroid levels are off. Please post your latest labs and a meds history so we can evaluate the adequacy of testing and current levels. Just being "in normal range" is seldom good enough to relieve symptoms. Please post reference ranges as well as they vary lab to lab and have to come from your own lab report.