By definition, culture is a particular form or stage of civilization, which also represent the behaviors and beliefs of people in the society. Institutions are well known organizations which have been established for a long time. In my opinion, I do believe that culture define institutions.
I would say that culture can be developed among a group of people without the existence of an institution. Our ancestors may be originated from some remote places long time ago where literacy was impossible. They might be hunters who lead a self-sufficient life, and generations could be carried on in this form of living. As they were more expertise in earning their lives by hunting or growing crops, they might start to have their own activities during their leisure time. It can be sewing, singing, dancing, creating their own language and these elements are the primitive part of a culture. Thus, they would have better communication with others and began to form institutions like council, school, and religious site. In other words, culture act as the base of institutions.
However, institutions also construct culture with reinforcement. For example, the leader of a religion can claim that the God forbid them to hunt deer as deer is a sacred animal, and eventually it will become the culture of the group of people to not hunt deer as food. This is how institution can shape culture in some extent but it does not produce a whole new culture. In conclusion, I think that culture defines institutions more than institutions define culture.