I'm sure everyone here has a story a little different from everyone else's; but we're here to find and give help about a common problem (and maybe commiserate a little bit too, LOL). I probably have one of the smallest breasts on the forum, but it (yes, "it," not "they," since I can grab a small handful only on one side) quickly gets sore and irritated if I go without some kind of bra(lette). When the problem started a few years ago, I didn't even realize yet that there was some breast tissue there. I thought it was just that for some reason the nipple was extra sensitive to friction on the inside of the shirt. I got the first sports bra with the intention of using it only when riding bike; but the problem progressed to where I needed something pretty much full time.
At first I was very scared to go out in public with it, but gradually realized no one is searching to see if I'm wearing a bra (although I haven't been in a group of teenagers with one). I still try to hide it completely, because most people are ignorant about gyne and our need for a bra of some sort, so the whispers and snickers and condemnation would start up, which is not good for anyone. I wish it were more-common knowledge. For me, being small, it's mainly about hiding the back; but for you, the same things that help hide the back might reduce the attention the front gets as well.
In addition to the obvious (like having the color closely match your skin's), something I find helps is wearing a starched, plaid cotton shirt over an undershirt. I'm using Marico's Revive instant starch, which is very easy and really helps hide the bumps and lines. Now in the winter I'm often using flannel shirts (with no starch). It still wouldn't pass the "hug test" as someone else here put it, so family gatherings might require wearing a bra that doesn't have the lumps in the back. Last summer I regularly wore a tank-type pullover bralette under striped polo shirts with collars, and I never saw any evidence that anyone noticed. I am however looking for a suitable bra with small, actual cups, as I think that would be more comfortable than a bralette that just smashes my breast down. They're rare in my size though.
Growing your hair and getting a close shave might work to keep others from realizing it's a man with breasts, but I can imagine other ways it would raise eyebrows. I for one would be extremely uncomfortable in the men's room if I thought that was a woman that just walked in! My wife and I were standing in the check-out line at the supermarket one time and there was a tall man in front of us dressed as a woman, and very convincingly so, except the height which is very unusual for a woman (and even for a man!), and the manly chin and hands. He was trying not to talk and give away that he was a man, but eventually was forced to talk a little bit to complete his business with the checker, and his bass voice was another giveaway. So, take everything into consideration. I grew up in another country, and kids there really made fun of me and the very few other kids who wore glasses, and in fact wouldn't admit when they needed glasses themselves even if they were obviously blind as a bat. Here in the States, kids wearing glasses is common, and no one gives it a second thought. I wish the acceptance of gynecomastia were that way.
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