Shortness of breath with hypothyroid? normal?

Yes: Hypothyroidism can be associated with shortness of breath, primarily through its effects on the heart. The heart rate can slow and the heart tissue can become weak; plus you can even get fluid build-up around the heart (pericardial effusion). But this is not "normal" and not a good sign.

Not normal: It is not normal obviously -- hypofunctioning thyroid is very damaging to the organ systems in the body -- so you need to get medical attention asap shortness of breath can be caused by heart and/or lung problems and not necessarily your hypothyroid -- again get medical attention now!