The three "Barricade Blaster" achievements are cumulative both through the Gaea campaign and the strategy game. To destroy a turret from direct control, you must be controlling either a Manta or Walrus of your own and shoot the final shot that destroys a turret controlled by the enemy AI. Any turrets that are destroyed by your carrier or vehicles that you are not directly controlling do not count for these achievements.
I would suggest completing the campaign first as you will need to destroy a lot of them to finish that. In a strategy game, invade an enemy island, starting by destroying any nearby Mantas, then save the game. Use a Manta of your own armed with missiles and an ammo box to destroy every turret in range, then reload the save. Rinse and repeat until you have 1000 kills, at which point the achievement will unlock. If the enemy controls Treasure, this is one of the best maps for this achievement. There are 13 turrets grouped around and just south of the Command Center, so this can go pretty quickly for a grinding achievement especially if you go in with missiles and a gatling.
If you choose to do this in the campaign, the first point at which you face enemy turrets and can destroy them is while invading Arachnid, the first island you attack after commandeering your carrier. There are several turrets you will have to destroy to succeed at this mission, so you can easily create a save file with your Walrus' gun pointed at a damaged turret.