Can a person in another country who is marrying a US citizen bring their 19 year old child with them?

Not likely. You must have married prior to the child's 18th birthday and then can file while the child is between 18-21. If you are not married, you can file a fiance visa for your fiance and then the child will come as a K2 dependent and file their green cards before 21. The fiance visa (K1) takes about 8 to 10 months to process.

There is only one way to do it and that is to go to the US as a fiancee, on a fiancee visa.  That way your 19 year old child will qualify for a dependent of a fiancee of a US citizen and once you are married, you and your 19 year old can both file for adjustment of status to permanent residence.  If you get married first and are petitioned by your US citizen spouse, your 19 year old child will NOT qualify for an immigrant visa because the marriage will have taken place after his/her 18th birthday.  But for the fiancee visa, the child is considered a child until he/she is 21.