Likes it been pointed out, its a turtle, so it isn't breathing the ammonia through its gills. But it is breathing.
Here I go with my lack of edumication, and extremely little experience with a turtle. As in, I came home from being out of town for a few days. And found a turtle in our indoor pond. My neice put it there. They hauled it off to IDK where a few weeks later.
IF ammonia fumes(vapors?) are heavier than air. He's going to be breathing ammonia.
Maybe someone will see this that knows, and chime in.
And / Or, stick your head in the tank and see if you feel like your breathing ammonia.
I know I've smelt some hampster cages(aquariums) before, and they can be pretty strong.
OK so lets assume hes breathing ammonia, and we would rather he not do that.
So....
Now our goals are to have nice air for the turtle and cycle the tank, to get rid of the cloudy water.
I'm going to guess that, being a turtle. 80-100% water changes won't bother it.
Going with that assumption. Heres what I'd do.
Change 80+% of the water.
This will leave plenty of ammonia in the water to keep working on the cycle.
You can clean out any mechanical filtration with water from the tank.
Do Not use tap water, it'll kill any bacteria you have aquired so far. Unless of course your on well water and know there isn't any chlorine/chlorimine in the water.
If thats the case. Use a desent temp of tap water.
Depending on whats in your tank for decor. Some bacteria will grow on that stuff too.
Basics:
Change water 2-3 times a week.
Clean mechanical filtration lightly.
Should have water clearing up in a week or two.
Happy you, and turtle! : )
P.S.
While all of this is going on. You could look into what affects a 5.0 pH water could/would have on a turtle, if any. I have my susspisions, that it could.