Leading out here, I feel that the responses you've received here are decent, however they're just throwing things out without identifying the what or why.
Turf algae is one of the most difficult to get rid of. Why? Because there really aren't many herbivores that actually consume it. First off, have you confirmed it's turf algae? It's difficult to tell from your photos but it does look like it. Is it growing in mats and very short? Never gets long or flowing? If so, then that's the first step to confirm.
Next, I'd recommend controlling your input, meaning what is going into the tank to ensure you limit what it uses to grow. What kind of lights are you running? Ensuring you're not throwing too much red and green lights will reduce the spectrum needed for algae growth. While your .03 phos levels appear to be good, it's likely masked due to the uptake of the turf algae. What are you feeding and how frequently? If you're feeding pellets or flakes, that's step one, stop and go to thawed and drained frozen so to minimize as much phosphate input as possible.
Export for not having a sump makes is going to reduce options and be more difficult but it's possible to manage. What is your water change process like and where are you getting the source water? Ensuring it's from a reliable source, even a local fish store can have suspect water. If you're getting mixed saltwater from a shop, check new/fresh saltwater before putting it in to see if there's any issues there. Clean water is means no additional nutrients for the algae to consume. Also maybe up your water change frequency for a while to keep those nutrients really low.
Lastly, natural control. Largely the only inhabitant that will touch that stuff is true turbo snails, even though they don't do all that great, but the Pithos crab should be your go to. They readily eat the stuff, albeit slow, but they do consume it.
With control of input, light spectrum, food and water quality. Export, water changes. And inhabitants, Pithos crabs, you should be able to work on getting rid of it.