The electrostatic tunability of graphene is vital in the field of active plasmons and would be beneficial in tunable infrared and terahertz optical element applications. The key to realizing broad tunability is achieving high carrier densities in graphene. Here we use an ion-gel, currently one of the most efficient dielectrics with ultra-high capacitance, to realize broadly tunable graphene plasmons (∼1270 cm(-1)) with low voltage modulation (∼4 V shifted from the Dirac point). We further explore the coupling between graphene plasmons and the molecular vibration modes of the ion-gel, since strong plasmon-phonon coupling can split the plasmon resonance peak into multi-peaks and reduce their tunability. Our experiments demonstrate weak plasmon-phonon coupling in the graphene/ion-gel system, which has limited effects on plasmon properties. These properties make ion-gels an effective dielectric for broadly tunable graphene plasmonic devices, such as new optical modulators, filters and wavelength multiplexers.