As Peachey says, there is no reason to use the past perfect : 'fast food <em>had</em> started to become more popular',  since we are in the 21st century now, and we are not referring back to an earlier period.

If you were to say  'by the turn [i.e. start] of the 21st century, fast food . . '  then you could use this form if you are writing a report.

It is better to start the sentence differently  e.g. 'In the twenty-first century . . ',  so the present perfect is appropriate.

We say "the twenty-first century", and we never use Roman numerals.

Because you're talking about the present situation (as a result of the past, or the progess from the past to now), you use present perfect.   I can't see any reason to use past perfect. 

No everyone ,like me never to touch