No one "allows" a child to be transgender. They are, or are not, transgender. And your personal disbelief that a child of three can have a clearly established gender identity is in direct conflict with observed facts. It is NORMAL for a child to have developed a fairly stable gender identity by ages 3 to 4. 

How Does Gender Identity Develop in Children?

Around two-years-old, children become conscious of the physical differences between boys and girls. Before their third birthday, most children are easily able to label themselves as either a boy or a girl. By age four, most children have a stable sense of their gender identity. During this same time of life, children learn gender role behavior—that is, do­ing "things that boys do" or "things that girls do.         - 

  - American Academy of Pediatrics: Gender Identity Development in Children