Green water in turtle tanks is often a result of algae, which in itself isn't particularly harmful. However, it's important to examine the conditions in the tank that can lead to an overgrowth of algae as that may indicate other issues that could be potentially harmful. Anytime you have water issues (cloudy or smelly water, or high ammonia or nitrate levels), the first things to assess are your tank size and filtration methods.
Conditions That Promote Algae
Algae tend to grow excessively in turtle tanks because the waste that turtles produce acts as nutrients for the algae. First, make sure the turtle is in a large enough tank―a tank that's too small can lead to waste products that are highly concentrated and increase algae growth. As a general rule, the tank should hold at least 10 gallons of water per inch of turtle—even larger is better. This means a full-grown turtle needs a tank larger than 100 gallons. If the tank is too small, water quality will be much harder to maintain, and the health and well-being of your turtle will be at risk.
Finding the Right Filter
Finding a very powerful filter is the next step, preferably a canister-type filter rated for a tank three to four times the size of your actual turtle tank. A pet store can help you choose one, as turtles are messy and need the extra filtration. Feeding the turtle in a separate container can also help reduce the amount of waste in the water that can lead to algae growth. When you clean the filter, don't disinfect it or use really hot water to clean it, because you don't want to kill the beneficial bacteria it harbors. The good bacteria is needed to degrade the turtles' waste products.
Make sure the lights on your turtle tank are not on for too long. The lights should be on for 10-12 hours per day; if they are on longer they may contribute to algae growth (not to mention stressing your turtle). Don't be tempted to cut the lighting too much―while the lights do contribute to algae growth, proper lighting (both basking and UVA/UVB) is critical to your turtle's health. However, if the tank receives any direct sunlight, moving it out of the sunlight can help significantly.
Keeping the tank clean is vital to water quality and will help with the algae—"Dilution is the solution to pollution," as the saying goes. Keep in mind there may always be some algae in the tank and perhaps even on your turtle's shell―trying to totally remove algae is futile and unnecessary. The goal is to keep the algae under control, but more importantly, make sure the water quality is good.
You can also add in animals that eat algae like snails and plecos. However, your turtle might eat them so you'll have to replace them continuously. Regular water changes will help remove the waste from the aquarium and take away the nutrients that algae need to grow. Be sure to use a gravel vacuum to clean the substrate of your turtle tank to prevent detritus build up.