Self-organized criticality (SOC) dynamics have been investigated in non-local transport regimes induced by the supersonic molecular beam injection in the HL-2A tokamak. Experimental evidence shows that the SOC or avalanche behaviors, such as the Hurst parameter, self-similarity and large-scale radial correlations in turbulence, are remarkably enhanced during the prompt non-local phase, together with an increase of inward propagation of turbulent events. These results highlight the importance of SOC paradigm during the transient non-local thermal transport in magnetically confined plasmas.