I will answer as simply as possible.
No. JF-17 Thunder did not shoot down jets, but it was part of the strike package which bombed Indian military installments. That's clear from this.
If you go back to some old threads where it was discussed on how the scenario played out (in Indian versions of 24 v 8 which were sham so I went on to explain why and what I think happened), you will find my comment to be the same: I said it from the very start, that it was more likely that the F-16 had the role of maintaining air superiority while the JF-17 and Mirage were involved in the strike.
As to why this was thrown around so much. Started off with people misinterpreting ISPR's statement "that no F-16 were used in the operation" as to meaning that there were no F-16 jets air borne. ISPR did clarify later that all jets were air borne, secondly his statement was correct as no F-16 was involved in the bombing, but was used to shoot down ingressing jets.
And then some people, specially media, threw around a lot of dirt with that, and specially people here and on twitter. Some did photoshop of killmarks (which was obvious, and many people here pointed the photoshop out, but some choose to ignore for some reason).
Why JF-17 Thunder didn't shoot any jet down? Very simple. That wasn't its mission and its loadout wasn't for that. It's job was to bomb sites next to Indian military depots and installments with Mirages, which it did successfully, and the F-16 jets were supposed to provide cover which they successfully did by shooting down a Mig and a Sukhoi.
It was a successful operation.
That's how the PAF planned, and that's how it went.