"Dadurch" means "therefore"
BUT "Dadurch, DASS" you would have to translate to "because" because of the grammatical differences.
And this would change the the original "dadurch, dass" to "weil".
This is the phenomenon we can experience when translating a text into another language and then translating it back (e.g. by using an online translator). Obviously the original meaning would have changed.
So in the end you'll find out that "dadurch, dass" turned out to have the same meaning as "weil" in your sentence.
Generally you can translate "Dadurch, dass" to "because", but "weil" cannot always be replaced with "dadurch, dass". Here are some examples why:
The usual case:
Ich konnte ihn retten, [weil] ich ihn festhielt [correct]
Ich konnte ihn retten, [dadurch, dass] ich ihn festhielt.[correct]
I could save him, [because] I held on to him. [correct]
Your example:
[Weil] sie die jüngste ist, hat sie viel mehr Freiheiten. [correct]
[Dadurch, dass] sie die jüngste ist, hat sie viel mehr Freiheiten. [correct]
[Because] she's the youngest, she has a lot more freedom. (uh...not sure about freedom, maybe liberties?) [correct]
Here's the problem:
Ich packe meine Koffer, [weil] ich morgen wegfahre. [correct]
I'm packing my bags, [because] I go on a trip tomorrow. [correct]