If there is chronic bleeding then understanding the source is very important. Without that the speculation can send you all over the place, as your prednisone pondering illustrates. For example, prednisone can cause peptic ulcers which in turn can bleed. The timing of your anemia may not be consistent with prednisone as a cause, but there’s also no good reason to expect the prednisone to have fixed it.
Your hematologist should be able to help you work out whether this is from bleeding, from red cell destruction, from failure of red cell production, or from a more obscure cause. Careful investigation that looks at (for example) the average age of the red blood cells can get pretty specific. Chronic disease can certainly cause anemia, in which case the challenge becomes figuring out what chronic disease is active in your body, and what you might do about it.