Sadly, MN is one of the most reliably transphobic places I have found on the internet, and I'm not surprised you've found yourself under a barrage of unhelpful messages
At this stage - I'd ask your child what they want to do, and keep an open mind. Perhaps they just wanted to share the thought and talk it over - in that case, be available and non-judgemental when it comes to talking it over - you might also want to consider professional counselling. A list of LGBT-friendly therapists can be found at www.pinktherapy.com/en-gb/findatherapist.aspx (NB one who thinks your child is mentally ill is going to be extremely dangerous). The Tavistock & Portman Clinic's waiting lists are exceedingly long, hence the suggestion of a private counsellor initially
There are plenty of things that your child could do that would help them to explore things privately - growing out hair, for instance (they can always cut it later, and lots of boys have long hair!) or having some feminine clothes that they wear in the house.
If your child seriously thinks they may want to transition, then take them to the GP pronto to get on the lengthy waiting lists for the Tavistock & Portman. If they decide they don't want medical intervention, then just ask to be taken off the waiting list. If they do want to consider medical intervention then they'll be nearer the top of the waiting list.
As someone who is very close to someone who has transitioned (went the other way, FTM. Initially thought he was a butch lesbian and then realised he was actually male and straight), and has been in most of the hospital appointments, I can assure you that this isn't something that happens without
a) very good reason - people only tend to transition when they can literally no longer live as their birth gender, even if they've had a good idea for a while that they're trans
b) without strong medical oversight - to even get to the stage of hormones, as an adult in his 20s, he had to see 2x psychiatrists and 1x endocrinologist, who all had to agree that this was the right pathway.
He's now living a perfectly good life - graduated from a top uni, qualified into his chosen profession, well paid respectable job, has a girlfriend etc. Transitioning hasn't stopped him from achieving any of his life goals. The hardest bit was his parents, who were thoroughly unsupportive and kept calling him by his birth name and female pronouns even though he had made it clear that this upset him.
There are some resources out there -
Mermaids is for people in exactly your situation www.mermaidsuk.org.uk/
GIRES was set up by the parents of a young transgirl many years ago www.gires.org.uk/
The main Trans Pride event is in Brighton this weekend if you live anywhere near. It's unlikely, from what you've said, that your child would be ready to go to such an event, but if you went alone (the park event is quite a nice, family festival / village fete atmosphere) then you would be able to speak to organisations like Mermaids face to face and also see that there are plenty of perfectly happy and successful trans people out there www.brighton-pride.org/transpride-brighton/
Do also be aware that there are people sometimes referred to as TERFs (trans exclusionary radical feminists) who choose to deny the very existence of trans people - I can assure you that this is not done for the benefit of trans people. They often like to refer to themselves as gender critical or transsceptics and other such terms - which are really just terms they've made up because transphobic sounds unpleasant. Nobody calls themselves sexuality critical or homosceptics. They'd just be homophobic.
Is your child going to want to transition in future? Who knows. All you can and should do at this stage is to be supportive and led by your child in terms of what they want to do. They won't be allowed to do anything irreversible without the approval of several doctors and a lengthy wait WinnieTheMe is quite correct in saying that a referral to a gender identity clinic is not necessarily a pathway to medical intervention. The GICs are quite overwhelmed with referrals as things are. They do not need to drum up new business.