Light aircraft pilot reported an NMAC in the traffic pattern at FFZ airport.

Narrative

I was cleared to enter the pattern on a downwind to runway 4L. I entered as instructed. I was told by tower to extend downwind. I did this as instructed. Tower told me to turn Base for runway 4L behind the aircraft in front of me on base. About 1-2 minutes later I was told to turn base by tower. This was about abeam the aircraft on base in front of me. I realized at this time that the aircraft I was following was actually landing 4R the runway abeam 4L. At this time; I began looking for the aircraft I was supposed to follow on final for 4L. I did not see them but was confident I would since tower cleared me to turn base. Tower contacted me soon after I turned final to see if I had traffic in site just then I saw the aircraft approximately 300 feet in front of me and 50 to 100 feet below me. I told tower I had traffic in site and needed to make a left 360 degree turn for spacing. Tower said negative and had the aircraft in front of me sidestep over to runway 4R. Both aircraft landed without incident.My aircraft was much faster than the one in front of me requiring me to get visual with the aircraft in this very busy traffic pattern earlier than I am used to.I believe that I should have queried the controller about the aircraft location or told tower I did not have traffic in site when I realized the traffic I thought I was following was not the correct traffic. Doing so would have aided me in seeing the aircraft sooner than I did helping me to maintain more separation.