Child is singular, and in this case, the word 'their' is singular also. Child ( like person, friend, student, teacher, neighbour and many other words) is non-gender specific. A child can be male or female, so to avoid the problem of whether to say 'his' or 'her', we use 'their' instead. We can also sometimes use 'they' or 'them' to refer to a person of unspecified gender. It's quite correct, and becoming increasingly common.

This is called 'singular they'. 'They' can be plural if the gender is unknown ('he' or 'she' would sound awkward, since the child could be either).