Hello! :D I apologize in advance if I explain anything you already know.
I've been doing a lot of research into this stuff recently, and the way the foster care system works in the US is both interesting and complicated. Emma really got a bad path through life. So I'll just go over briefly what we know about her childhood, based on what we've seen/heard and how things work in the US
When Pinocchio went to the authorities with baby Emma, attempts would have been made to find her parents. while that was happening, both she and Pinocchio would have been in a foster home (which they were, as we saw in "The Stranger") at some point, that effort would be abandoned and Emma, as a baby, would be adopted. (Which she was, per her statement in "Pilot")
Emma tells us in the pilot that when she was three, her adopted family had their "own kid" and sent her back. So she's back in the foster system :(
Emma makes statements about foster kids being "meal tickets" and getting shuffled around. This was obviously her experience. Add in the fact that she frequently ran away, and we can conclude that Emma was moved around quite a bit, even transferred from state to state.
The last family we see her with, in the timeline, is Ingrid, not the family you were mentioning in the forum thread. Ingrid was working on the paperwork to adopt Emma, but that fell apart. That's the last time we see her in any sort of foster care or family.
The age of majority in the US is 18 (or 19 in two states). However, Emma makes a statement in "True North" about being in the foster system for 16 years. That's the minimum age (except in California) that a child can sue to become emancipated, or a legal adult. This is what Emma did. It's further supported by the fact that she was arrested at 17 and treated as an adult, and was allowed to independently make the choice to have her son adopted.
I know that's a lot of detail just to say, "she didn't have an adopted family when she was a teen", but like I said, I'm super interested in this whole thing right now XD