'their exhibitions' is correct and can be used in written English in many contexts.
Example sentence: Many people come to see their exhibitions of fine art.
A timeline leading into the galleries details the activities of nonconformist artists, who exhibited in private apartments or had their exhibitions shut down by the government.
(The Internet of course aids greatly in the process; many galleries lavishly document their exhibitions).
However ephemeral their exhibitions, nonprofit galleries still make a critical contribution to the art market.
Traditionally, museums have had a sort of wary, protectionist attitude about their exhibitions.
Traditionally, museum executives have a wary, protectionist attitude about their exhibitions.
Every year he produced pictures for their exhibitions which were sought after by galleries.
We like exhibitions still to be on when the magazine comes out but students often make their exhibitions too short".
Punk bands perform at nightclubs, guerrilla filmmaking is not uncommon, and experimental painters have toured their exhibitions abroad.
