At present, there are many deblending algorithms to deal with the separation of the blended data. However, the convergence speed for most algorithms is relatively slow, which is difficult to implement in industrial production. Thus, we propose an accelerated joint iterative method (AJIM) to separate the blended data based on the iterative shrinkage thresholding algorithm (ISTA) and the acceleration linearized Bregman method (ALBM) in the curvelet transform domain. In conventional ISTA, the updated deblended result of each iteration is obtained by thresholding curvelet coefficients and the gradient term. It is stable but always needs a lot of iterations to obtain a good performance. ALBM is proposed based on LBM via an acceleration factor at each iteration, and the acceleration factor can increase the proportion of unthresholded curvelet coefficient, which further accelerates the convergence. This method converges more faster than LBM and ISTA at the beginning, but it can cause artifacts at the late iterations because some unthresholded coefficients contain some wrong information. Based on advantages of ISTA and ALBM, the weight parameters with linear and exponential functions are proposed to control the contribution of ISTA and ALBM in the proposed AJIM. Therefore, AJIM can greatly speed up the deblending efficiency at the beginning and improve the deblending accuracy in the late stage. At last, AJIM can obtain the same deblending effect as ISTA, but only a few iterations are used. Synthetic and field data tests are used to demonstrate that AJIM-based deblending can effectively and quickly separate the blended data.