The main thing you need to consider is that children's hair is very different from adult hair. We are born with very fine, soft hair that slowly changes in texture as we grow up. In general, children maintain their 'baby hair' until the age of around six to nine years. After this, their hair changes to thicker adult hair that remains the same until graying, baldness, or both. In your child's case, perhaps it's a little early to expect thicker growth at the young age of three. Let nature take its own course and she should have an adequate head of hair in a few years.
You have also mentioned frequent tonsuring; while this can make certain types of adult hair growth back thicker and denser, it does not work with children's hair. Your family's vegetation diet is not a problem, as long as you ensure that your child gets al the major food groups, essential nutrients, and vitamins. Take care that your diet has important mineral like as zinc, iron, and magnesium, and vitamins such as A, B-complex, C, and E. These are important in the hair growth process, as are 'good fatty acids' called omega-3 and omega-6 fatty acids.
Meanwhile, it is always a good idea to introduce your child to a sensible hair care regime at an early age. In a year or two, you can even let her manage the routine of shampooing, rinsing, and brushing her own hair. For now, you can stay in charge, but let her know what you are doing, and why. You are probably already using a mild baby shampoo; add a few drops of olive oil to provide additional nourishment, protection, and body to the hair. This can also help to reduce tangling of hair, which is very common with children's finer hair. You can also put a few drops of tea tree oil solution as preventative measure against hair lice, which is common in younger school-going children. Remember that children do not need daily shampoos, so use them only when the hair is really messy. Allow the hair to dry naturally, since any hurried drying can tangle up the hair. While the hair is still a little wet, you can gently rub in a few drops of vinegar, which is an excellent natural conditioner. Never use an adult brush on a child's hair, since their scalps are tender. Get a wide tooth comb that will not entangle the hair and will help to keep it manageable.
Warning: The reader of this article should exercise all precautionary measures while following instructions on the home remedies from this article. Avoid using any of these products if you are allergic to it. The responsibility lies with the reader and not with the site or the writer.