Many studies and institutions such as the American Medical Association and the American Psychiatric Association have acknowledged that gender dysphoria is highly distressing and that it can be dangerous if left untreated.
Transgender kids are generally at much higher risks of self harm, anxiety, depression and suicide, but a 2016 study showed that transgender teenagers who are supported in their transition are no more depressed and only slightly more anxious than cisgender peers.
Some people argue that the risks associated with medically transitioning are greater than those associated with untreated gender dysphoria. However, the risks are quite low in comparison to developing depression, anxiety, and self harm or suicidal tendencies. If the minor chooses to take puberty blockers, the only possible health risk is that the minor could have lower than average bone density. Yet, doctors who prescribe puberty blockers are acutely aware of this possible risk and regularly check bone density while the child/adolescent is taking the medication. Even though their bone density could be lower than average, this deficit is not linked with any severe health risks. This is also reversible; when the medication is stopped (whether that be because the individual no longer identifies as transgender or because they are switching to hormone replacement therapy), the individual will regain normal bone density after a few years.
Gender-affirming hormone therapies, like taking estrogen or testosterone, are generally only prescribed to patients who are at least 16 years old. Gender-affirming hormones can change an individual's body in ways that are not easily reversed, such as changes in body hair growth and body fat redistribution. Some of these changes may be completely irreversible. However, one should keep in mind that these changes are cosmetic. The changes do not pose any health risks. Thus, as long as the individual is sure that gender-affirming hormones is what they want, there aren't really any associated risks. Making sure that the individual is definitely transgender and that hormones are definitely what they want can be done through extensive counseling, and most doctors will not even prescribe hormones unless the individual has been "insistent, persistent, and consistent" about their gender identity for at least six months. There is the chance that the individual could change their mind about their gender in the future, but in that case they can take measures to mitigate the cosmetic changes caused by hormone therapy. 
Other kinds of medical transition are rare for minors, and genital surgeries are not offered at all to non-adults. Overall, the types of transition procedures that are available to minors are very safe, especially compared to leaving gender dysphoria untreated.