R' Chaim Kanievsky pointed out that there is an explicit Gemara that you can cut a child's hair before the age of 3.
The Gemara in Moed Katan 14b discusses whether a child can get a haircut on Chol Hamoed. The Gemara says that not only a child who was born on Yom Tov or Chol Hamoed can get a haircut but even other children. You see explicitly that if a child is born with a lot of hair you are allowed to cut it. This is quoted l'halacha in Shulchan Aruch Siman תקל"א.
Here is the background to the story. Someone asked R' Chaim the following question. His son is only 2.5 and he wants to take him to Meron to cut his hair on Lag Baomer (he won't be 3 yet). Should he be machmir to wait until his son is 3 to cut his hair? R' Chaim answered that it is explicitly permitted to cut a child's hair before the age of 3.
IMHO this question is a perfect example of what is wrong with the Charedi world today. Not cutting a child's hair before the age of 3 is a minhag, if you don't have the minhag then you don't have the minhag. To ask whether you should be machmir is ameratzus. What kind of chumra is it to wait until 3 to cut a child's hair?
It may make sense to be machmir when you have a a situation where we posken that something is mutar but some poskim say assur. Therefore to avoid violating an issur according to some shitos you are machmir. This depends on the issue and the person etc. However, by a minhag like this of not cutting hair, how can someone think it is שייך to be מחמיר?