As a trans women, I feel obligated to provide my perspective.
This is an incredibly complex issue, and I will not be able to discuss all of the different aspects of it. While I do think it's important to understand that the experience of every trans person is different, generally speaking, being trans is not like being gay, lesbian, or bi. There is a lot more going on.
(There are some trans people who don't suffer from gender dysphoria, but for the purposes of this comment, I'm going to sidestep them and the issues they raise) Most people know that your brain doesn't finish developing until you are about 25 years old. To the average person, it stands to reason that the decision to undergo Hormone Replacement Therapy (HRT) is not one that you should be making unless you can appropriately understand the long-term consequences. Since you can't really do this until your brain finishes developing, it follows logically that you should delay the decision to undergo HRT until this age.
But unfortunately, time is working against trans people in this regard, because the physical effects of puberty have long since made their mark by this age.
What may seem like wonderful, normal, or perhaps mildly annoying traits for cis people can be a source of crippling, incurable stress, anxiety, and/or depression (more often referred to as 'gender dysphoria') for trans people. This can, and frequently does, lead to suicide for trans people.
But this treatment, and its availability, can have unintended consequences. The social repercussions of delaying puberty can be very serious (teenagers can be really mean), and unsupportive parents can use puberty blockers to buy more time for them to try to convince or coerce their children into taking a path that they will later regret. Providing children with unbiased information about gender dysphoria is vital in that it builds confidence in trans children who are not confused or in denial about their gender dysphoria, which helps them get the treatment they need by standing up to adult figures who would claim to know better and suggest alternative treatments.
But it is also vital to helping children who would otherwise fail to recognize their symptoms. Information about gender dysphoria needs to be provided repeatedly, both early on and later when children are on the verge of starting puberty, because confidence is a hell of a thing that can take a lot to build up, and denial is a hell of a thing that can take a lot to overcome. Trans people, and their parents/guardians, can struggle to come to terms with the fact that their gender dysphoria is a symptom of a disorder that can only be treated with HRT. This is especially true in light of the fact that gender dysphoria is something that virtually always worsens with age; This is why you see older people transitioning instead of simply continuing to cope in other ways. I say this as someone who had OBVIOUS dysphoria as a child and teenager, but failed to recognize it until the age of 22. I was able to do this only as a result of the information I found online through /r/asktransgender on Reddit. I suffered many consequences as a result of my denial, although I should count myself lucky in that I did not commit suicide despite a frequently overwhelming desire to do so. Many trans people are not so lucky. Tl;dr: Puberty can be DEADLY for trans children, but puberty blockers should be a last resort for children who are still unsure of their gender identity after repeatedly receiving information about gender dysphoria. Parents/guardians and doctors should not casually consider a treatment that undermines the feelings and social development of children that are already struggling to comes to terms with an extremely difficult decision.
It's important to realize just how devastating this can be for some trans people.
Biological females can develop wide hips & breasts, and will have a smaller rib cage. Biological males can develop broad shoulders & facial hair, and will have a larger rib cage. These traits will be finished developing long before the brain does.
Some of these effects of puberty cannot be reversed without undergoing extremely expensive, and potentially unsafe medical procedures.
These procedures are considered "cosmetic" by virtually all insurance companies, which means you must pay for them out-of-pocket. Many trans people cannot afford them.
Other effects cannot be reversed at all with current medical technology.
As coffeesp00ns mentioned, puberty blockers can delay these effects to give the child more time to seriously introspect and weigh their feelings against the reality of transitioning.