Rechargeable Zn–air batteries (ZABs) have received extensive attention, while their real applications are highly restricted by the slow kinetics of the oxygen reduction and oxygen evolution reactions (ORR/OER). Herein, we report a “bridge” structured flexible self-supporting bifunctional oxygen electrode (CNT@Co-CNFF50-900) with strong active and stable Co-N/C@pyridine N/C@CNTs reaction centers. Benefiting from the electron distribution optimization and the advantages of hierarchical catalytic design, the CNT@Co-CNFF50-900 electrode had superior ORR/OER activity with a small potential gap (ΔE) of 0.74 V. Reinforced by highly graphitized carbon and the “π–π” bond, the free-standing CNT@Co-CNFF50-900 electrode exhibited outstanding catalytic stability with only 36 mV attenuation. Impressively, the CNT@Co-CNFF50-900-based liquid ZAB showed a high power density of 371 mW cm-2, a high energy density of 894 Wh kg-1, and a long cycling life of over 130 h. The assembled quasi-solid-state ZAB also demonstrated a high power density, attaining 81 mW cm-2, with excellent charge–discharge durability beyond 100 h and extremely high flexibility under the multi-angle application. This study provides an effective electrospinning solution for integrating high-efficiency electrocatalysts and electrodes for energy storage and conversion devices.