We investigate thermo-electronic behaviors of flexible AlGaN/GaN heterostructure field-effect transistors (HFETs) for high-power operation of the devices using Raman thermometry, infrared imaging, and current-voltage characteristics. A large negative differential conductance observed in HFETs on polymeric flexible substrates is confirmed to originate from the decreasing mobility of the two-dimensional electron gas channel caused by the self-heating effect. We develop high-power transistors by suppressing the negative differential conductance in the flexible HFETs using chemical lift-off and modified Ti/Au/In metal bonding processes with copper (Cu) tapes for high thermal conductivity and low thermal interfacial resistance in the flexible hybrid structures. Among different flexible HFETs, the ID of the HFETs on Cu with Ni/Au/In structures decreases only by 11.3% with increasing drain bias from the peak current to the current at VDS = 20 V, which is close to that of the HFETs on Si (9.6%), solving the problem of previous flexible AlGaN/GaN transistors.