So this is normal even for a partial water change? It was about 40% water change.
The cloudiness has nothing to do with a water change. It's because when you did the water change you added a water conditioner which removed the chlorine/ chloramine. Chlorine + chloramine are added to your tap water to kill off bacteria and other organisms that would be harmful to you if you drank it. Chlorine and chloramine are harmful to fish, and so you need to remove this from your tap water before you add it your aquarium. I'm not 100% sure why your fish have survived so far, maybe you don't have much Chlorine or chloramine in your tap water, maybe the fish that are surviving are somehow more tolerant, but regardless Chlorine and chloramine will be burning your fishes gills, and won't be doing them any good.
There are numerous bacteria found in aquariums, and so far they have been killed by the chlorine/ chloramine. There are bacteria you don't really want, such as the ones causing your bacterial bloom.
There are bacteria you want to keep alive though. The nitrogen cycle functions by microbes consuming the ammonia that your fish produces. The chlorine/ chloramine in your tap water was killing those too, so your cycle never established.
The bacteria you are seeing are feeding off the excess nutrients (ammonia that is present because your cycle hasn't established). As your cycle establishes, the bacteria responsible for the nitrogen cycle will outcompete the bacteria causing your bloom, and the bloom will go away. This isn't something you should see with every water change, but it might take a couple of months for your cycle to establish and it clear up.
So just to be clear, API Quick Start doesn't eliminate chlorine/chloramines?
No. API Quickstart doesn't remove chlorine/ chloramine. It's a supplement that they claim contains those microbes responsible for the nitrogen cycle and helps to cycle the aquarium. In reality, these products simply don't do anything useful, and even if they did, adding them to chlorinated water would just cause anything useful to die, same as any other microbial life.
I had been using API's master kit and those results were all in "nominal" ranges. Algae growth and plant growth were pretty steady.
Unless those "nominal" ranges are zero for ammonia and nitrite, then they are present. Low levels might be relatively safe for fish, but they are still there and are still food for the bacterial bloom you are now seeing. Same for the algae and plants. They will love the nutrients present from an uncycled tank.
In a cycled aquarium you should be seeing zero ammonia and nitrite, and your nitrate should be steadily rising between water changes. If you are seeing any ammonia or nitrite in your water then you arent cycled. The nitrogen cycle should be turning all the ammonia and nitrite into nitrate.
What precisely are your water parameters?