This is actually pretty confusing for people who understand it and have researched it a lot, so let me try to explain:

Usually when a person finds out that they are transgender, they often and understandably want to start presenting as the other gender. Not everyone who is transgender has a problem with their genitals specifically though, meaning that there are transgender people who could go their whole lives without gender reassignment surgery and be totally fine with it assuming they were allowed to live as their preferred gender socially.

Transgenderism isn't inherently distressing. What causes the most distress and suicidal tendencies in transgender individuals tends to be the result of social stigma, bullying, threats of violence, etc. Studies have shown this, but I don't have them on hand unfortunately. So this is where things get tricky: gender dysphoria refers to the amount of distress someone has about the fact that their body doesn't match their preferred gender. The mismatch is not inherently distressing to everyone who experiences it, which is why everyone who is transgender is not inherently diagnosed with gender dysphoria. Some people's only reservations about being transgender are how they'll be accepted and treated socially.

The transgender wikipedia page actually has quite a bit of information on this:

Transgender people may meet the criteria for a diagnosis of gender identity disorder (GID) "only if [being transgender] causes distress or disability."[77] This distress is referred to as gender dysphoria and may manifest as depression or inability to work and form healthy relationships with others. This diagnosis is often misinterpreted as implying that transgender people suffer from GID; this misinterpretation has greatly confused transgender people and those who seek to either criticize or affirm them. Transgender people who are comfortable with their gender and whose gender is not directly causing inner frustration or impairing their functioning do not suffer from GID.