We can put it this way, I want you to do it/ It's better that you do it.
this sounds like a foreign speaker (such as spanish) directly translating what is correct grammar for them into english. quiero que tu vayas a la tienda - correct in spanish, but translated literally into english it says " i want that you go to the store" while we say "i want you to go to the store."
in my opinion, they are two different things.
to want plus infinitive is the correct for, obviously.
But to say *I want that she undertands me* hasn't got the same meaning of * I want you to understand me*.
The first sounds more a command, the second a desire.
Lastly, what about *I want that she's happy* from the australian band The Middle East?
It is no court evidence, sure, but it goes to show it is in use.
I believe this is correct because it is an example of the present subjunctive in English:
However, as a native speaker of English, this sounds very unnatural. I recommend that one say:
I want you to do it.
It's grammatically false but as people use this form more often it should be linguistically introduced in grammar because people's opinion counts.