We investigate incoherent broadband cavity enhanced absorption spectroscopy using a tailored supercontinuum source. By tailoring the supercontinuum spectrum to match the high reflectivity bandwidth of the mirrors, we achieve an unprecedented spectral brightness of more than 7 dBm/nm at wavelengths where the effective absorption path length in the cavity exceeds 40 km. We demonstrate the potential of the source in spectrally broadband measurement of weak overtone transitions of carbon dioxide and methane in the near-infrared 1590 nm - 1700 nm range and evaluate its performance against that of a typical superluminescent diode source. Minimum detectable absorption coefficients (3σ) of 2.2 × 10(-9) cm(-1) and 6.2 × 10(-9) cm(-1) are obtained with the supercontinuum and the superluminescent diode sources, respectively. We further develop a spectral fitting method based on differential optical absorption spectroscopy to fully and properly account for the combined effect of absorption line saturation and limited spectral resolution of the detection. The method allows to cope with high dynamic range of absorption features typical of real-world multi-component measurements.