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It is possible to bring your fiancé here for the marriage. Starting this way, the ONLY appropriate way to do this is to apply for the K-1 fiancé visa.
This process involves filing through USCIS, waiting the four to six months for approval, and then your fiancé must process through the U.S. embassy or consular post in the home country (this can take another one to two months normally, in some cases almost double that).
The fiancé visa petition is something of a “hybrid” – named as a nonimmigrant visa, largely viewed as a nonimmigrant visa, yet it is much like the Immigrant Visa for Alien Relative filed after marriage and in large part turns into this petition after the fiancé comes to the U.S. and marries the U.S. citizen. It requires that all of the same things be proven: valid relationship (here, this includes that the couple has met in person at least once in the two years leading up to the petition, or that in the culture involved it is normal not to), financial support, etc.
Because in your situation, this would be committing fraud.
It isn’t unusual for someone to enter as a visitor planning only to visit, or in some other status intending to do only what that status permits, and after some time here circumstances change and people develop different feelings, goals and plans. In such a case, nothing dishonest was done and a marriage-based green card would be approvable.
However, the visitor visa requires “nonimmigrant intent” – the intent to visit briefly and then return to a home abroad. This is NOT the intent if the plan is to come to the U.S. to marry a U.S. citizen fiancé and then apply for permanent residence. Representing that it is would be a misrepresentation, and could cause denial of the case later. There is a presumption of fraud if one enters as a visitor and then does something inconsistent with the required intent within a certain period after entry.