"To carry the weight of the world on one's shoulders" indicates an emotional state, and even suggests that person does this unnecessarily. For your sentence, use "to take on the duty of...": "After the death of his parents, he took on the duty of supporting his two brothers." If you tried to use the idiom of "carrying the weight of the world", you'd actually be saying that he physically carried both brothers.

"carry the weight of the world" is an idiom. It means that the person is burdened with worries and issues. So you could not really decompose it to mean supporting his two brothers. However if you did say that, an English speaker would understand that the person has taken on the responsibility of looking after his brothers and it is not an easy task.