I don't know when the words transman and transwoman were first used. Not that long ago, surely? They certainly didn't enter the mainstream until 10 years ago at the very most. I wish there had been a lot more kickback when they were first introduced, as they are massively confusing words, and I have to assume that was the intention.
In English, words that end in -woman normally refer to female humans. A lot of those words are very old-fashioned or obsolete now - washerwoman and policewoman, for example - but the odd one is still used, e.g. Frenchwoman, countrywoman. Transwoman was therefore either coined by someone who had no grasp of how language works, or it must have been designed to make people think that it referred to a female person.
There was a very recent survey that showed that a lot of people do indeed think that transwoman is the word for a female human who identifies as a male. And of course if you think that and hear people talking about whether transwomen should be in women's changing rooms, prisons, refuges, sports etc you will probably think 'What's the problem? No matter how a woman identifies, she belongs in the women-only spaces and services, not the men-only ones, for reasons of her own safety and that of other women, and for privacy and dignity.'
However, transwoman was coined as a word for a male human, i.e. a man, who identifies as a woman. This can't be repeated often enough, along with the vitally important connected fact that the vast majority of transwomen don't have genital surgery, so have a fully functioning penis and testicles.
It remains mystifying to me why these small but vociferous groups and their allies have been allowed to dictate to the rest of us how we should talk about them. They would probably say this is like people with disabilities rejecting terms that have been used in the past as insults. However, those aren't the only groups who've asked to be able to choose their own names, and the rest of us don't always agree to do it. Paedophiles, for example, are keen to rebrand themselves as minor attracted persons or MAPs. Most of us don't go along with that.