I've heard the phrase in a movie: "I don't want to see that happen" and i'm a bit confused with the form of "happen" in this context. I've though that the correct form is infinitive: "I don't want that to happen" or " I don't want to see that happened". Can someone make it clear? Many thanks in advance!
Your confusion on this point is understandable. Speaking not as a grammar expert, but just as an English speaker...
Construction 1: "I don't want that to happen" is correct to say.
Construction 2: "I don't want to see that happen" means the same thing as construction #1 but has a softer feel, like you would feel bad if it happened, not just that you are opposed to it.
It is parallel in construction to the following examples. In each pair the two phrases mean basically the same thing (different flavor, same basic meaning).
"I don't want that to happen" and "I don't want to see " "I don't want you to go" and "I don't want to see " "I want him to stay" and "I want to see "
The common pattern here is the part in in each example is a phrase describing what you want to see. In each case "to see" could mean literally to watch it happen but more likely it means "to see it come to pass".
But also consider these two examples which mean the same thing but one uses "to happen" and the other uses "happen":
"I don't want to allow that to happen" "I don't want to let that happen"
Construction 3: "It would be unfortunate if that happened". This also means something similar to the first two constructions but correctly uses "happened".
Construction 4: "I don't want to see that happened" is not correct English. You could say "I don't want to see that that happened" which is correct but very awkward. Or perhaps "I don't want to see that it happened" but this still feels a bit awkward or odd to me.
I realize this is not a technical explanation of the grammatical reasons but perhaps it illustrates some common patterns that may be helpful.