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

I think, under the certain circumstance of the sentence, there's subtle difference of the meaning of the translation... in my opinion: by using "弄明白", the author want to emphasize that things happened in very short time, that 他the unprepared Mr.Dursley didn't get to know what he saw, so he glance back...in order to see again what he saw before... and by using "明白", things would be subtly different, it could mean that Mr.Dursley did see what happened moment before, just he don't understand the thing (could give the reader the feeling that it's the thing's too complicated to be understood)... in the sentence, plus the rare part of it "he glanced back", I think it's better to use "弄明白"... *** it's all personal opinion, hope it helps or at least not misguiding ^^ Have fun !

怎么通顺就怎么表达，没太大的区别，例如：你弄明白了吗和你明白了吗都一个意思。

我觉得楼上说的挺好的 。。强调的重点不一样。。弄明白强调的是动作，是过程。。。 你的问题和搞清楚一样，你清楚了吗？ 你搞清楚了吗？这两句应该也会经常用到的吧

mostly is the same ,one is v+n ,one is n or v