I feel like they are both correct in the speaking but also feel like maybe one of them is incorrect grammatically because of the obeject pronoun. Is there a rule to object pronoun before or after verbs? I looked for it in the internet but all the articles are super especific and I can't understand it very well.
Both sentences are grammatically correct, but they mean different things.
They wanted me to fit in this world.
That means that they wanted it to be true that I would fit in this world.
They wanted to fit me in this world.
That means that they wanted to take steps to make me fit in this world. Of course, that also means they wanted me to fit.
(In that example, the word "into" would be a better choice, because it suggests the transition implied by the meaning of the sentence.)
When someone fits in somewhere, they make successful efforts to be acceptable, compatible, etc.
When I joined the team, I tried to fit in by learning everyone's names and bringing doughnuts.
When another person fits me in, or other people fit me in somewhere, they make efforts or arrangements so that there is a place for me.
We wanted Joan in our team, but the office was full, so we moved Peter nearer to window so we could get another desk and fit her in.