You can only have 3 Active Bounties at one time and only Active Bounties count their progress, however, you can accept lots of bounties and return to the Bounty Broker and switch them as you please to make things a lot faster.
For example, there is a bounty that requires you to use 8 fuel cans, so simply pull up to the Broker when you need to refuel, turn on that bounty and fill ‘er up! When you’re done, switch out the bounty and move on. This will let you progress the more passive bounties as you need whilst keeping Active Bounties as things that will be most useful when out in the field.
Look to stack bounties that have similar requirements like bladed kills, combat kills etc. This will also make progress on this otherwise tedious achievement much faster. Bounties also carry between communities in the base game, so you can make progress in various places and even use not turned in Bounties as a method of starting with some free guns in a new community.
Note: a number of users reporting tracking issues with this achievement in August 2020. Seemingly relating to the most recent patch given that no prior issues were noted.