A wave-band-tunable optical fiber broadband orbital angular momentum (OAM) mode converter based on a helical long-period fiber grating (HLPFG) and dispersion turning point (DTP) tuning technique is demonstrated both theoretically and experimentally. The DTP tuning is achieved by thinning the optical fiber during the HLPFG inscription. As a proof of concept, the DTP wavelength of the LP1,5 mode is successfully tuned from the original ∼2.4 µm to ∼2.0 µm and ∼1.7 µm. With the help of the HLPFG, broadband OAM mode conversion (LP0,1→LP1,5) is demonstrated near the 2.0 µm and 1.7 µm wave bands. This work addresses a longstanding problem that the broadband mode conversion is limited by the intrinsic DTP wavelength of the modes and provides a new, to the best of our knowledge, alternative for broadband OAM mode conversion at the desired wave bands.