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According to my understanding of this,
I think these are just based on the grammatical aspect
like "弄" is a verb, which means to make someone.....something
"明白" is also a verb, but compared with "弄", theres only a little difference.
弄明白 is to make someone understand/believe
as for 明白, we use it for like "Cool, I understand(我明白了)."
but it could actually be "Cool I (have made myself) understand.(我弄明白了)"
it depends on the way you say,
more specifically, "弄明白" can also be something like "I have to figure this machine out! (我必須弄明白這個機器)" or ( I have to figure out how to use this machine. 我必須弄明白如何使用這個機器 )
"明白/弄明白"
They usually appear on novels, plays, or dramas.
In real life, when we wanna say that "I get it./ I understand./ I see."
Its more common to use "我懂了/我知道了" because "我明白了" is sorta literary.
These are just my own opinions, hopefully they could help :)
Best,
Amy
