SMPS design problem, low available current
Using a bridge rectifier In the output instead of a fast diode is missing the point of what a flyback converter does and partially ruins performance. Then, most bridge rectifiers have awful reverse recovery time and will kill any decent performance when switching at high frequencies.

These are two massive issues that caused your problem. The next is lack of snubber network on your primary winding. Circuits not using a snubber can get away without one because they are low power designs less than a few watts but, you are wanting maybe 60 watts. No flyback snubber will likely harm the MOSFET or degrade it significantly.

Finally, the driver chip is not “man” enough to control a MOSFET that is capable of delivering anything like 60 watts to the output. You need to rethink this in my opinion.