To me, there are three major external conflicts in this book.
First, I think there is a conflict between Kit and society in general when she moves to Connecticut from Barbados. She finds that things are much different in her new home, as seen when she finds out that swimming is associated with witchcraft.
Second, there is the conflict between Kit and Hannah on the one side and the Puritans on the other. This is connected to the first conflict.
The last conflict I see is between the colonists and the Crown as the colonists oppose the attempts to change their charter.
I think the main external conflict in the book is the conflict that exists between Kit and the Puritan society in general. Kit comes to Connecticut from Barbados where she was used to a certain level of wealth and prestige. Kit was also used to an easier life in which she did not have to do much manual labor. Her life in Barbados stands in sharp contrast to the Puritan society that she finds herself in for much of the book. The Puritan women are used to working and working hard. They wear drab clothing as well. For much of the book, Kit is simply a "fish out of water." Her attitude and expectations create conflict for her with other members of the town.
The division that exists between Kit and the town only grows as Kit begins to spend more time with Hannah Tupper. As the fever strikes the town, Hannah and Kit are targeted as necessary scapegoats. Kit is accused of being a witch, and she is put on trial. The accusations placed against Kit are not solid; they are rooted in the fact that Kit is different than the rest of them and spends time with other people the town has alienated.
Another conflict that is present in the story is the conflict that exists between the colonists and England. The best example of this conflict is the charter the colonists have that grants them certain rights and a certain amount of freedom. During this time in history, England attempted to revoke colonial charters. The townspeople in the story do not want that, so in chapter 15 they steal and hide the charter. The purpose of hiding the charter was to prevent it from being revoked so that Connecticut could maintain its own rule. This kind of conflict between the colonists and England will continue to grow until a war is necessary for the colonists to achieve full independence from England.