Bathroom exhaust vent into crawlspace
No, at least not under the international residential code (IRC):

M1501.1 Outdoor Discharge
The air removed by every mechanical exhaust system shall be discharged to
the outdoors in accordance with Section M1506.3. Air shall not be
exhausted into an attic, soffit, ridge vent or crawl space.

And more specifically:

M1507.2 Recirculation of Air
Exhaust air from bathrooms and toilet rooms shall not be recirculated within a residence or to another dwelling unit and shall be exhausted directly to the outdoors. Exhaust air from bathrooms and toilet rooms shall not discharge into an attic, crawl space or other areas inside the building.

The reasoning is simple: the exhausted air is likely to contain a lot of moisture, and will therefore result in mold growth in your crawlspace. Your local codes may differ depending on where you live, but it's still not a good idea.