psx coverting files to iso

I am currently using ePSXe which works perfectly fine and I have some iso games that work. Using imgburn on my original games work really well but if I download one it comes has either .bin or .bin and .cue. Now this is the funny part....
I've tried to mount it has a virtual drive but none will work so I tried to burn one to a disc using imgburn and it works from the disc. I used imgburn again to make an iso and that worked too. Obviously I don't want to keep doing this method so I was wondering if I can cut the middle man and turn it into an iso? I have tried using imgburn and powerISO but the file won't load. Any suggestions are much appreciated.

UPDATE
Noticed one site, the downloads actually play but the other does not. Not sure why but at least it's a step in the right direction.

No it does not work that way, ISO and BIN are not really convertible, but you could decompress-recompress. I would not do that. Just try running the .bin file with the "Run ISO" option.