There are actually two large conflics going on simultaneously in Chapters 12 - 16. The primary conflict is an internal one that is central to the book, concerning what Winnie should do about her newfound knowledge about the magic spring. The secondary conflict is an external one, having to do with the actions of the man in the yellow suit who will go to any lengths to gain possession of the land housing the magic spring.
In Chapters 12 and 14, the Tucks talk to Winnie, trying to get her to see their point of view on the magic spring. In Chapter 12, Pa Tuck himself speaks to Winnie in a passionate manner, explaining just why it would be dangerous for word of the spring's magic properties to get around to the general public. Pa Tuck believes that the idea of living forever sounds very appealing at first, but that once it is obtained, it is not at all the wonderful gift it appeared to be at first glance. He describes the despair of being left out of the natural stream of life, and says that by the time people who take advantage of the water's magic powers realize the situation they are in, it will be too late to go back and make things right. Pa Tuck sees the spring as a dangerous thing, and wants to impress upon Winnie the importance of keeping it secret.
In Chapter 14, Jesse presents another side of the dilemma to Winnie. He agrees with his father that the spring should be kept secret, but adds that as long as they all know about its magic powers, they might as well take advantage of it. Jesse wants Winnie to wait until she is the same age as he is - seventeen - and then drink some of the water so that she will be stuck at that age forever. At that time they could perhaps get married, and enjoy eternal life on earth together. Winnie does not know what she should do about her knowledge about the magic spring. Her conflict is internal; only she herself can decide what she is going to do.
The second important conflict occurring simultaneously to all this concerns the actions of the man in the yellow suit. He knows that Winnie has been kidnapped by the Tucks, and goes to Winnie's family, offering to secure her release if they will sell him their land, the land with the magic spring. This conflict is external, as the Fosters are being blackmailed into selling their land by the conniving and opportunistic man in the yellow suit.