A primary design decision in HTTP/2, the successor of HTTP/1.1, is object multiplexing. While multiplexing improves web performance in many scenarios, it still has several drawbacks due to complex cross-layer interactions. In this paper, we propose a novel multiplexing architecture called TM3 that overcomes many of these limitations. TM3 strategically leverages multiple concurrent multiplexing pipes in a transparent manner, and eliminates various types of head-of-line blocking that can severely impact user experience. TM3 works beyond HTTP over TCP and applies to a wide range of application and transport protocols. Extensive evaluations on LTE and wired networks show that TM3 substantially improves performance e.g., reduces web page load time by an average of 24% compared to SPDY, which is the basis for HTTP/2. For lossy links and concurrent transfers, the improvements are more pronounced: compared to SPDY, TM3 achieves up to 42% of average PLT reduction under losses and up to 90% if concurrent transfers exist.