The intrusiveness of Web tracking and the increasing invasiveness of digital advertising have raised serious concerns regarding user privacy and Web usability, leading a substantial chunk of the populace to adopt ad-blocking technologies in recent years. The problem with these technologies, however, is that they are extremely limited and radical in their approach, and they completely disregard the underlying economic model of the Web, in which users get content free in return for allowing advertisers to show them ads. Nowadays, with around 200 million people regularly using such tools, said economic model is in danger. In this article, we investigate an Internet technology that targets users who are not, in general, against advertising, accept the trade-off that comes with the “free” content, but—for privacy concerns—they wish to exert fine-grained control over tracking. Our working assumption is that some categories of web pages (e.g., related to health or religion) are more privacy-sensitive to users than others (e.g., about education or science). Capitalizing on this, we propose a technology that allows users to specify the categories of web pages that are privacy-sensitive to them and block the trackers present on such web pages only. As tracking is prevented by blocking network connections of third-party domains, we avoid not only tracking but also third-party ads. Since users continue receiving ads on those web pages that belong to non-sensitive categories, our approach may provide a better point of operation within the trade-off between user privacy and the Web economy. To test the appropriateness and feasibility of our solution, we implemented it as a Web-browser plug-in, which is currently available for Google Chrome and Mozilla Firefox. Experimental results from the collected data of 746 users during one year show that only 16.25% of ads are blocked by our tool, which seems to indicate that the economic impact of the ad-blocking exerted by privacy-sensitive users could be significantly reduced.