Overconfidence is a prevalent problem and it is particularly consequential in its relation with scientific knowledge: being unaware of one's own ignorance can affect behaviours and threaten public policies and health. However, it is not clear how confidence varies with knowledge. Here, we examine four large surveys, spanning 30 years in Europe and the United States and propose a new confidence metric. This metric does not rely on self-reporting or peer comparison, operationalizing (over)confidence as the tendency to give incorrect answers rather than 'don't know' responses to questions on scientific facts. We find a nonlinear relationship between knowledge and confidence, with overconfidence (the confidence gap) peaking at intermediate levels of actual scientific knowledge. These high-confidence/intermediate-knowledge groups also display the least positive attitudes towards science. These results differ from current models and, by identifying specific audiences, can help inform science communication strategies.