The Bristol City Museum and Art Gallery event, which attracted nearly 4,000 people a day, and the new Saatchi gallery were the only British entries. A Buddhist exhibition at the Tokyo National Museum topped the list, with nearly 16,000 visitors a day.
The Louvre in Paris topped the museum attendance list, with more than 8m visitors, the Art Newspaper reports.
The British Museum was second with 5.56m visitors.
Both the Banksy and the Saatchi exhibition, which examined Chinese contemporary art, were free to the public.
The Bristol City Museum and Art Gallery has never featured in the list before.
The attendance for last summer's Banksy exhibition, which was not publicised in advance because it had to be prepared in secret, beat the likes of the Anish Kapoor show at London's Royal Academy of Arts.
Japanese events took the top four places in the exhibitions league.
Along with the Buddhist exhibition based on the treasures from the Kohfukuji temple, the Tokyo National Museum also hosted Treasures of the Imperial Collections, which was fourth on the list.
In second place was the 61st Annual Exhibition of Shoso-in Treasures at the Nara National Museum, in the ancient Japanese city of Nara.
A show based on 17th Century painting from the Louvre at Tokyo's National Museum of Western Art, featuring works by the likes of Rembrandt, Velazquez and Poussin, was in fourth spot.
In the museums list, three other British institutions made the top 30 besides the British Museum - the National Gallery, Tate Modern and the Victoria and Albert Museum, all in London.
The effects of the recession have yet to show themselves as big exhibitions can take years to prepare and may have had sponsorship deals in place for a long time. But with fewer businesses able to sponsor blockbuster exhibitions, visitor numbers may be down in next year's list, according to the Art Newspaper.
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