"I want you to do it" seems far more natural to me in English. Both "I want that you do it" and "I wish that you do it" both sound off to me but I'm not sure how to explain it yet.
You could say, "I wish that you would do it" in English, but it would probably have to be imperfect subjunctive in Spanish. "Quisiera que lo hiciera." or "Me gustaría que lo hiciera."
There is a big difference between "I want that you would do it" and "I want you to do it." The latter is a strong expression of a desire for a person to get with it and do something. The former is a weaker expression being presented largely as a declaration of a hope or wish more than anything.