<p>Incompatibility is a fundamental feature of the quantum world, with the incompatibility of observables famously captured by the Heisenberg uncertainty principle. As a matter of fact, the quantum channel incompatibility (QCI) renders a more valuable insight and wider scope into the incompatibility, in the light that the incompatibility essentially originates from the disturbance of the eigenstates imposed by a quantum channel. However, characterizing QCI remains challenging since it is intractable to apply two channels in alternative orders simultaneously on the same quantum system. A recent theoretical proposal [Phys. Rev. Lett. 130, 170201 (2023)] introduced a general definition of QCI based on measurement processes and showed that a quantum switch (QS), an implementation of indefinite causal order (ICO), can enable direct estimation of QCI. The quantum switch, already known for its applications in quantum communication, metrology, and quantum thermodynamics, offers a powerful tool for probing such process-level incompatibilities. In this work, we experimentally measure the incompatibility of several quantum noise channels, including depolarizing, bit-flip, and phase-flip channels, using a Sagnac-type quantum switch. Furthermore, we demonstrate that estimating QCI via ICO provides substantial improvements in resource efficiency over traditional process tomography approaches. Our results demonstrate that the quantum switch not only offers quantum advantages in information processing but also serves as a versatile platform for investigating fundamental aspects of quantum mechanics.</p>