そのうちに、自分を直してくれた医者のように、「人の怪我や病気を直したい。」と、考えるようになったのです。
Some day, like the doctor who cured him, he came to think “I want to cure people from injuries and illness.”
その頃、日本はアメリカとの戦争の真っ只中にありました。
At that time, Japan was in the midst of a war with the United States.
治虫がその時住んでいた町にも、毎日のように爆弾が落とされました。
The town where Osamu lived then was bombed almost every day.
戦争が終わったとき、治虫は十六歳でした。
When the war was over, Osamu was sixteen years old.
そのうちに、自分を治してくれた医者のように、『人の怪我や病気を治したい。』と、考えるようになったのです。
Before long he was treating himself like a doctor, and Osamu started thinking that he wanted to heal people injuries and illnesses.
そのころ、日本はアメリカとの戦争のまっただ中にありました。
At that time, Japan was in the middle of the war with America.
治虫がその時住んでいた町にも、毎日のように爆弾が落とされました。
Almost everyday bombs were dropped on the town where Osamu was living at that time.
戦争が終わった時、治虫は十六歳でした。
When the war ended, Osamu was sixteen year old.
In the first sentence:
そのうちに、自分を治してくれた医者のように、『人の怪我や病気を治したい。』と、考えるようになったのです。
I was struggling with the 医者のように part. After checking deepl, the translation is like a doctor who treated him. Kind of obvious, but that was not at all how I read it myself.
If he treated himself, you wouldn’t use てくれた, which means that he is receiving something (i.e. the treatment). The one who provides the treatment is the 医者.
自分を治してくれた qualifies the doctor as the one who cures Osamu.
I like that you share your mistakes, even after you noticed them, so that others can learn from them.
Did the exact same thing, in hindsight the を and くれた made it obvious. But yeah, went a bit too fast.
たくさんの人たちが死んでいくのを間近い見た治虫は、生き延びたことを、心からありがたいと思いました。
Seeing many people dying so close to him, Osamu felt sincerely grateful that he had survived.
「もう、戦争は終わったんだ。世の中を明るくする、夢のある面白い漫画を描いていこう。」
“Now, the war is coming to an end. I will draw entertaining manga with a dream to cheer society up.”
医者になるための勉強をしながらも、治虫は漫画を描き続けていました。
Osamu continued to draw manga while he was also studying to become a doctor.
やっぱり、漫画を描くことが大好きだったからです。
That was because he still loved drawing manga.
治虫は、早速漫画の売り込みを始め、4コマ漫画の連載が決まりました。
Osamu’s manga quickly began to sell heavily, and a 4-panel manga series was decided.
たくさんの人たちが死んでいくのを間近に見た治虫は、生き延びたことを、心からありがたいと思います。
Osamu, who saw close-up many people dying, thought that he was grateful from his heart to have survived.
「もう、戦争は終わったんだ。
“Now the war has ended.
世の中を明るくする、夢のある面白い漫画を書いていこう。」
To make the world bright, I will continue drawing dreamy, amusing comics.”
医者になるための勉強をしんがらも、治虫は漫画を書き続けていました。
While he also studied to become a doctor, Osamu continued drawing comics.
やっぱり、漫画を描くことが大好きだったからです。
In the end, it was because he loved drawing comics.
治虫は、早速漫画の売り込みを始め、四コマ漫画の連載が決まりました。
Osamu immediately started promoting his manga, and a 4-panel comic serialization was decided on.
The use of 夢のある in this sentence confuses me:
世の中を明るくする、夢のある面白い漫画を書いていこう。
Is it referring to a dream of Osamu’s, or is it modifying 漫画 somehow?
a “dream-having amusing” manga
In addition to 200kanji’s translation, let me deconstruct the Japanese a bit:
夢のある面白い漫画
Here you see two things that qualify 漫画: 夢のある and 面白い.
I think the second one (面白い) should be clear: It’s just an adjective qualifying a noun, resulting in “interesting manga”.
The first one (夢のある) might look a bit confusing. Here it may help to know that this is just a short and simple phrase: 夢がある “a dream exists” where が is turned into の. This is a special case which can only be applied in relative clauses to help clarify the scope of the subject of the relative clause. (see also Relative Clauses | Japanese with Anime).
So all together this is a manga that is interesting and where a dream exists, so to speak.
Thanks Nicole. And my understanding was that “the dream that exists” is Osamu’s dream. i.e., the manga that was drawn was interesting (I actually went for “funny” rather than “interesting”) and it was his dream (to create it).
I want to add two things: I think the dream here is not Osamu‘s dream but there is a dream in his manga. This sounds weird but I guess it‘s about the mangas giving hope to the people, it‘s about a vision for the future, not only about Osamu‘s personal aspirations.
I dont think „to make the world bright“ or „to cheer the world up“ is what the texts says, because the 世の中を明るくする-part is modifying 漫画 here, so he wants to create mangas, that make the world/society bright/happy. But maybe this difference is not that big. 
十七歳の、学生漫画家の誕生です。
It’s the birth of a seventeen-year-old student manga artist.
十七歳の、学生漫画家の誕生です。
Seventeen years old student-mangaka was born.
十七歳の、学生漫画家の誕生です。
It was the birth of a 17 year old student manga author.
そのうちに、長いストーリー漫画の注文も来るようになりました。
Before long, orders for long story manga began to come in.
治虫の漫画を読んだ人は皆、口々に言いました。
Everyone who read Osamu’s manga said unanimously:
「こんな漫画は、今までに見たことがない。スピードと迫力があるね。」
“I have never seen a manga like this. There is speed and intensity.”
治虫は、漫画と同じくらい映画も好きでした。
Osamu liked movies as much as manga.
まるで、映画を見ているような動きのある画面を、紙の上に描いて見せたのが、治虫の漫画の新しさだったのです。
The novelty in Osamu’s manga was that he showed scenes drawn on paper, with movements seen quite as in a movie.
そのうちに、長いストーリ漫画の注文もくるようになりました。
Before long, a longer story comic also came to be ordered.
治虫の漫画を読んだ人はみな、口々に言いました。「こんな漫画は、今までに見たことがない。スピードとはく力がるね。」
Everyone read Osamu’s comic, and each and every person said, “I’ve never seen this kind of manga until now. It has speed and impact, right?”
治虫は、漫画と同じくらい映画も好きでした。
Osamu also liked movies about the same as manga.
まるで、映画を見ているような動きのある画面を、紙の上に書いて見せたのが、治虫の漫画の新しさだったのです。
He conveyed on paper movements just like those seen in movies on screen, which was the novelty of his manga.
That last sentence was rough. I think I got all the parts right in the end, but I had to double-check my work on deepl.
EDIT: Also, I might be the only one in the manga fandom who uses “manga” and “comics” interchangeably, without even thinking about it. I just love the whole medium 
I think みせた here is an auxiliary verb and should be written in hiragana. Osamu managed to draw scenes on paper that have movements like a movie.
I’m not sure but I don’t think the text says that everyone read the comics. The みな refers to everybody having the same positive opinion.
In my opinion 画面 here is not about the movie screen. It’s the scenes he is drawing because it’s the direct object connected to 描いてみせた.
まるで、映画を見ているような動きのある画面を、紙の上に書いて見せたのが、治虫の漫画の新しさだったのです。
What was novel with Osamu’s manga was that he made sure to draw scenes upon paper quite similar to the movement seen in films.
Translating is hard… I felt like I understood the general meaning of the above sentence upon first reading but as soon as I try and use English to capture that meaning, I’m copying and pasting clauses, moving them before and after each other, and tweaking all their connections.
To me it’s always helpful to start at the end of the sentence and then move backwards.
治虫の漫画の新しさだったのです
The novelty of Osaus manga was
書いて見せたのが
that he managed to draw
紙の上に
on paper
画面を
scenes
動きのある
that have movement
まるで、映画を見ているような
similar to watching a movie