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A nonlocal fracture criterion with accounting of the work during bonds deformation at the fracture process zone has been implemented analytically for analysis of bridged cracks growth. This criterion consists of two conditions: (1) the necessary energy condition of the crack tip limit equilibrium, which takes into account the energy release rate to the crack tip and the rate of deformation energy consumed by bonds in the crack bridged zone; (2) the sufficient condition is the equality of the crack opening at the bridged zone trailing edge to the bond limit stretching. Subcritical and quasi-statical regimes of bridged cracks growth have been formulated on these fracture conditions. Regimes of bridged cracks growth are analyzed in detail for the case of an internal straight bridged crack in homogeneous material with bonds traction, which is constant and independent of the external loading. Analytical expressions are obtained for the deformation energy release rate and for the rate of deformation energy consumed by bonds. The main fracture parameters, the critical external load and the crack bridged zone size in the limit equilibrium state are determined and analyzed. The limit cases of a crack which is filled with bonds and a crack with a small-scale bridged zone are considered. A comparative study with the well-known force fracture criterion for bridged crack growth is performed.