US citizens can sponsor their fiancee for a visa to the U.S. The visa is known as a K1 visa and requires that the couple get married within 90 days of the alien's arrival in the U.S. The alien can then petition to adjust status to that of lawful permanent residence as the spouse of a U.S. citizen.
Federal immigration law allows for the minor child of an immigrating fiancee to accompany their parent to the U.S. on a K2 visa. The minor's ability to remain in the country long term and to obtain LPR status is also dependent upon the fiancee actually marrying the person who filed the K1 visa. The regulations define a child for K2 purposes as a son or daughter under the age of 21. If the child is over 21, they cannot accompany their mother or father.
One important thing to keep in mind is that then the fiancee and her child go to the embassy for the visa interview, the parent will have to demonstrate that the child's other parent consents to the child moving to the U.S.