Trimming and Hair Growth
A: No. Throughout the years, the theory that cutting the hair promotes the growth of thicker hair has been proven false, regardless of the location of the hair being cut, or the length or methods of cutting. Hair grows according to its genetic instruction encoded in the individual's cell structure.
In children, the hair may grow faster or slower, depending on growth cycles and genetic programming. At some points a child's hair may seem to grow at an accelerated rate, while at other times the hair seems not to grow at all. Rapid growth periods may coincide with growth spurts in the body, and/or the onset of puberty and the hormonal changes that take place at that time.
In addition to an increase in the rate of growth, many individuals notice changes in the texture and wave pattern of the hair at these times as well, since not only does an individual mature, but the follicles of the hair mature as well.
So, the bottom line is this: if you want to change the style of your child's hair to one that is easier to manage for you as a parent, feel free. However, don't expect there to be any increase in the rate of growth as a result.
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See also:
Hair myths
Will horse shampoo help hair to grow longer and thicker?
Can cutting hair determine how thick it grows back?
Will Biotin help to thicken hair?