Hello folks I am an Americian citizens. engaged to be married to a citizen of Sweden. Together we have a child, that was born in Sweden, but has americian citizenship. We have been together both living in America and now Sweden for the past 7 years. We are coming home to America august 1st. My fiancee will come into the country as a tourist for her granted 90 days. I want to know if I can marry her while she is in the US and be allowed to keep her past the 90 day time limit. What should I do??

Your fiance should not attempt to enter the US on a tourist visa in this manner. You have a child together, and she is in a long-time relationship with a US citizen. Clearly, she is entering without the intention to depart.

Further, your fiance would need to obtain a B-2 visa, rather than enter on the visa waiver. This would require that she be interviewed (under current DOS policy). If she did not disclose her relationship with you and her USC child, BCIS could conclude that she made a misrepresentation to get her visa. This could result in her exclusion.

A safer course for both of you is to either get married and file an I-130 for her, or file a fiance petition. Both of these will take a long time, but it will prevent you from having any problems with BCIS or the Consulate on fraud issues.

A riskier path, but safer than a tourist visa, is for your fiance to apply for a student visa. After she has been in the US on F-1 status for more than 60 days, you can get married and move forward on a green card application. This would require finding a program she could enroll in, but will probably be faster than filing for a fiance visa or getting married and filing an I-130.

It will be better for you to get married there and sponsor your spouse at American Consulate and also apply for K-3. It will be much easier. The other option is that you should apply for Fiance Visa from the United States, but that will take about one year for your spouse to come here. Under first option, your spouse will be here soon and if your spouse will come on K-3 then will be eligible for EAD, and will be allowed to adjust Status in the United States.
Good luck.

How can k3 be any faster when both it and k1 are dependent on I-129f?

K-1 is a fiance visa petition. K-1 Visa can not be issued unless Fiance Petition I-129F is approved. K-3 Visa is issued to spouses of USC while their petition I-130 is pending, so that spouse of a USC can join the other spouse in the United States while all other paper work is pending. Thereafter, spouse is allowed to Adjust Status in the United States instead of home country.
Good luck.