Among her prizes were best female artist, best streaming song for Shake It Off and the ceremony's top honour, artist of the year.
British star Sam Smith won best new artist - and accepted his trophy using flash cards after having vocal surgery.
But Kanye West was booed as he closed the show with a medley of hits.
Audience members jeered as the rapper's sisters-in-law, Kendall and Kylie Jenner, introduced his set; and the booing continued during his performance.
Surrounded by pyrotechnics and largely in the shadows, West's delivery on All Day and Black Skinhead was angry and intense. And while some disapproved, other fans could be seen pumping their fists to the tracks.
TV audiences were also left mystified as the first minute of his set was censored by broadcasters ABC - reminiscent of West's performance at the Brit Awards earlier this year.
Other artists were more favourably received, with Britney Spears and Iggy Azalea receiving a standing ovation for the first performance of their new song, Pretty Girls.
-BBC