Subwavelength channels filled with near-zero-index (NZI) media can realize extraordinary optical functionalities, for example, tunneling electromagnetic wave without reflections, but usually confined in a narrow wavelength band due to the material singularity (refractive index n≈0), which seriously limits the practical potentials. In this Letter, we show this limit can be fundamentally overcome by an alternative, named near-zero-index-featured (NZIF) structure, with the singularity transmuted via a controlled optical conformal mapping, enabling the device implementation with nonmagnetic normal dielectrics (i.e., relative permittivity >1). Their equivalence is strictly examined through a subwavelength tunneling waveguide. Classic wave tunneling features in a broad frequency range are revealed in various confined geometries. These properties are robust against the disturbance of several kinds of structural defects benefited from the infinite effective local wavelength. The broadband and lossless NZIF medium proposed here provides a promising way to pursue the fascinating light controlling functionalities as initially enabled by singular NZI materials.