The DSM-5 versus popular misconceptions on transgenderism and gender dysphoria

That being transgender is a mental illness.

The DSM-5 does not agree. The problem, according to the DSM-5, is not the gender identity or transgenderism "per se," but the distress that may accompany it in the form of gender dysphoria. From the p. 451 of the DSM-5 in the section on "Gender Dysphoria" (emphases in original):

Gender dysphoria refers to the distress that may accompany the incongruence between one's experienced or expressed gender and one's assigned gender. Although not all individuals will experience distress as a result of such incongruence, many are distressed if the desired physical interventions by means of hormones and/or surgery are not available. The current term is more descriptive than the previous DSM-IV term gender identity disorder and focuses on dysphoria as the clinical problem, not identity per se.

That believing you are of another gender means you are delusional.

 This is not true either. In the differential diagnosis of gender dysphoria with "[s]chizophrenia and other psychotic disorders" (p. 458):

In schizophrenia, there may rarely be delusions of belonging to some other gender. In the absence of psychotic symptoms, insistence by an individual with gender dysphoria that he or she is of some other gender is not considered a delusion. Schizophrenia (or other psychotic disorders) and gender dysphoria may co-occur.

That gender dysphoria is just like body dysmorphic disorder and should be treated as such.

The DSM-5 lists these as distinct conditions, and provides assistance for the differential diagnosis between the two (p. 458):

An individual with body dysmorphic disorder focuses on the alteration or removal of a specific body part because it is perceived as abnormally formed, not because it represents a repudiated assigned gender. When an individual's presentation meets criteria both for gender dysphoria and body dysmorphic disorder, both diagnoses can be given. Individuals wishing to have a healthy limb amputated (termed by some body integrity identity disorder) because it makes them feel more "complete" usually do not wish to change gender, but rather desire to live as an amputee or a disabled person.