1. A system, comprising: one or more computing devices configured to: receive, at an edge node of a content delivery network (CDN) service, an encapsulated representation of a browsing request generated at a client browsing device, wherein the client browsing device implements a client-side component of a split-browser service, and wherein the split-browser service includes one or more server-side browsing engines of a provider network; determine, at the edge node, whether the browsing request is to be extracted from the encapsulated representation for at least an initial stage of content retrieval analysis at the edge node; in response to a determination that the browsing request is not to be extracted for the initial stage of content retrieval analysis at the edge node, identify, at the edge node, using a routing knowledge base available to the CDN service, at least a portion of a network route to a selected server-side browsing engine; transmit, from the edge node via the at least a portion of the network route, the encapsulated representation of the browsing request to the selected server-side browsing engine; extract, at the selected server-side browsing engine, the browsing request from the encapsulated representation; and initiate, based at least in part on a configuration setting associated with the client browsing device, one or more content retrieval operations from the selected server-side browsing engine to respond to the browsing request.