So how is it possible there is current?
Mobile electrons are the (plentiful) majority charge carriers in the N type region whilst mobile holes are the (plentiful) majority carriers in the P type region
The fact that the holes that diffuse to the N type region recombine there does not take away from the fact that they crossed the depletion region contributing to the total current through that region.
Similarly, electrons that diffuse to the P type region recombine there but, as with the holes, they crossed the depletion region and contributed to the total current through that region.
Since the device is overall charge neutral, the (total) current through the depletion region is the current through the device (don't forget that there is an attached external circuit through which the current circulates).
Because they recombine, doesn't that form a depletion region in
I don't follow your reasoning; the depletion region forms at the junction on the, e.g., N type side since electrons diffuse to the P-type side leaving behind ions.
But when an external circuit is connected with the appropriate polarity, electrons flow from the external circuit into the N-type side which reduces the width of the depletion region.