A time-dependent nonlinear equation for a nonstationary curved flame front of an arbitrary expansion coefficient is derived under the assumptions of a small but finite flame thickness and weak nonlinearity. On the basis of the derived equation, stability of two-dimensional curved stationary flames propagating in tubes with ideally adiabatic and slip walls is studied. The stability analysis shows that curved stationary flames become unstable for sufficiently wide tubes. The obtained stability limits are in a good agreement with the results of numerical simulations of flame dynamics and with semiqualitative stability analysis of curved stationary flames. Possible outcomes of the obtained instability at the nonlinear stage are discussed. The instability may result in extra wrinkles at a flame front close to the stability limits and in self-turbulization of the flame far from the limits. The self-turbulization can also be interpreted as a fractal structure. The fractal dimension of a flame front and velocity of a self-turbulized flame are evaluated.