When we cycle a new tank bacteria colonize it and consume the ammonia and nitrite the fish produce. The end produce from the bacteria is nitrate. There are bacteria that consume nitrate but they require very low oxygen levels to live. Those conditions will not exist in an aquarium. So all aquariums will have nitrate although in some cases at levels so low your typical test kit will not detect. Ammonia and nitrite are also present in all cycled tanks but typically at levels low enough that many test kits will show zero.
There are typically only 2 ways to remove nitrate, cycle water out occupationally. You are doing a 50% water change weekly which is correct. Water changes also remove organics The second common way to reduce it is to have health plant growth. All plant and animal tissue contains nitrogen. Fish will get is through the food. But for the plants they must remove it from the water. So if you have very good plant growth your test kit might only see a trace of nitrate. Plants will also absorb other forms of nitrogen (ammonia, nitrite, and urea). The last way to remove nitrate is to use a special nitrogen absorbent in your filter canister.
Plant additives (= fertilizers) also add organics
While some fertilizers do contain organics many do not. However most fertilizer contain nitrate because it is one of nutrients plants need most. Those fertilizers that contain Organics typically use it to keep iron in solution. Iron is a trace nutrient that is not include in some fertilizers.Since iron is a trace nutrient only very low levels or organics are typically found in a bottle of plant fertilizer.
since Nitrate is a very powerful acid it is typically not found in the by itself. It always will react with something in the water creating a nitrate salt. Common nitrate salts found in aquariums are potassium nitrate, sodium nitrate, calcium nitrate, and magnesium nitrate. You might even find Ammonia Nitrate in aquarium water (ammonia is a base and will react with nitrate) Since nitrate is a salt it will typically not evaporate from the aquarium like ammonia, nitrite and nitrogen can. Nitrate will just build up in an aquarium until it is mechanically remove in a water change or absorbed by a plant.
So keeping organics low clearly solves or helps to prevent many problems.
But sometimes it can be a mystery. I have 8 tanks in my fish room, and in my 90g I have had slightly hazy water resulting from some sort of diatom/bacterial bloom for almost two years. It comes and goes.
Sudden water changes are frequently written off as a mystery simply because people do not do fully test the water for all nutrients in the water. To do a full water chemistry workup you would need to test for about 50 different materials in the water. Most people just don't have the time or money to do all of these tests. so often the cause of a sudden algae issue never gets identified.
Many people don't run water hardness GH or total Alkalinity tests on there water. After all those levels are stable in the tap water and therefore they should stay stable in an aquarium. This thinking is not correct because tap water is not chemically active. An aquarium in comparison is always having chemicals removed and added. So a aquarium is chemically very active place. I have Seen GH changes dure to and or poor plant growth, and skipped water changes and when I started to get a high phosphate levels (caused by copper deficiency) under control my KH took a major drop due to plant growth suddenly accelerating. I have also seen iodine levels suddenly climb a couple of times and I think that was due to a drop in potassium levels ( I don't have potassium test kit but at the time i was not adding it to my RO water).
So your sudden change in nitrate may have had many causes. Which one is hard to identify. But since you are having a algae issue and i believe you have a planed tank. You might be deficient in some nutrient and a reduction in plant growth could explain a sudden increase in nitrate. You might want to try adding some Seachem Flourish Comprehensive fertilizer. It is a complete fertilizer that has everything plants need. Other causes of low plant growth is not enough light, or inconstant aeration of the t water.