In case of forward bias, p-type of the diode is connected to the positive terminal of the battery and n-type is connected to the negative terminal. When load is applied, negative terminal of the battery repels the electrons in the n side of the diode and they start moving towards the p-side.The more you apply voltage, more the width of depletion region is reduced and barrier potential reduces. At the same time, positive terminal also attracts the electrons moving from n side to p side. Once you cross the barrier potential(0.7 V in case of Si), current starts to flow in the opposite direction of the electrons. Here, current flows because of majority charge carriers( electrons in n-side and holes in p-side)