It's very moving to see a mother doing the right thing, trying to understand their child rather than just rejecting them. Rest assured that you're doing the right thing, and that you're acting in your child's best interest, potentially avoiding many years of trauma or even suicide! I share Lady Smith's view, as well as the opinion of everyone here.
There's a section named
Significant Others in the forums, you might want to check it out, as it has many people in a situation similar to yours one way or another.
Susan's Place's
wiki also has a lot of great information available.
This web resource has many interesting links specifically aimed at significant others, too, but some might not be directly related to your child's case.
As Ms. Grace pointed out, it's common to have a very early understanding of one's gender identity. It's not a norm, as in not everyone sees it so clearly from such an early age: for me, for example, I knew
something about my gender was very wrong since I was around 5, but feared too much that trying to understand myself better would lead to strong rejection from the people around me. Because of that avoidance, I only really understood I was transgender at age 16.
Since your child is very young, you might want to exercise caution for a while, as they need more time to understand themselves better. There's a strong possibility your child is not actually a girl from what you said, but you can't be sure of anything in such a short time span. A gender specialist (as in not someone who will "heal" your child, but who'll help with understanding) would be very helpful if you have the money for it and depending on where you live, but otherwise your best bet is researching and asking for help, which is exactly what you're doing right now.
Careful with researching online, of course. The internet is a very powerful tool, but there are many strongly biased opinions about the subject. Be especially careful with media reports, as those tend to be sensationalist more often than not. Talking to other transgender people and asking for information and/or helpful links from then might be your best resort online, in fact. No "doctor" knows nearly as much as they think they do about transgender issues, because not only have they never been through any of it, they often fail to consider our perspective at all.
(E.g.: most pediatrists will tell you that it's most likely only a phase, most psychologists would try to "heal" your child out of it, etc
)
Take special note of if your child has any behavior that'd be expected only of a trans boy (a boy that feels trapped in the body of a girl), rather than of a boyish girl. For example, they might play "peeing while standing up" with a water gun or something similar when they get a little older. It's a random example, I just mentioned it since I personally know a trans man who used to occasionally play in such a way!
Let your child be happy as they wish. If anything, if they really turn out to be transgender, puberty and the subsequent body changes could bring them GREAT suffering and heavily complicate their life later on. This is entirely avoidable, as it's possible both to delay puberty itself until your child has time to decide and switch your child's puberty with a male one (irreversibly so, meaning you two very much need the time to decide). Both are done with a specialist's aid.
Delaying puberty in itself could bring some risks, such as possible infertility. It's nothing that'll bring nearly as much complication and suffering to your child's happiness as letting they have their puberty developmental changes if they are transgender, but it's something to consider.
All in all, both of you most likely have a very rough time ahead of you. People are not only very badly informed and closed-minded, they also often refuse to accept any information or insight you offer at all, instead choosing to shove whatever they take as truth down other people's throats.
But do go on. If your child is truly transgender, they'll never be happy living as anyone but themselves. And God certainly didn't make anyone just so that they could wear a mask and live by lying to everyone the rest of their lives.
It is a great, great blessing to your child to have a mother such as you.
PS: I was really surprised about your sister-in-law as well, her information seems great for the most part (which is rare)
If you ask me, if a kid ever "outgrew" gender dysphoria then it was most likely not gender dysphoria at all, it's not something you can conveniently change your gender identity for just because it brings you suffering. But just the part where your sister-in-law tells you that it's fine for your kid to be who they are... I wish society was more like that.
