This one can be tricky in online play because you need to get all the turret kills yourself instead of minions or teammates getting the kills. The easiest way to do this will be to do a local "Bot Match" and set the AI difficulty to 0/5. I made sure my other teammates were local controllers to ensure that they would not get the turret kills. After that you just need to defend both lanes while gaining enough solar to unlock your highest level upgrades. Lonestar can be helpful here by upgrading his blaster and his dynamite to do a lot of damage to turrets.
One note is to be careful that your minions don't get the turret kills so try let them die regularly and take down the turret's last health by yourself (or using the bull as a shield) to ensure that last hit.
Edit: I forgot to mention that not all maps have 4 turrets so you're best off doing the first map Ribbit IV. This is probably the most straightforward map with two turrets on the top and two on the bottom.