Turning in crucible bounties doesn't progress past 4
That quest was created when Shaxx only offered normal daily bounties. With Shadowkeep Shaxx now offers in addition to those regular daily bounties two weekly bounties and a repeatable bounty for bright dust. The weeklies and repeatable bounties don't count because they didn't exist when that quest was created. Prior to Shadowkeep everyone working on that step of the quest was only able to progress by doing the 5 bounties a day.