1) Property assessment is ONLY based on a few key data (location, story of the house, square footage, and etc). Hence assessment value can be deviated from its true market value.
2) Banks can't do appraisals. Bank can order appraisals, which in turn are carried out by some local independent companies.
3) If appraisal is significantly lower than county assessment value, one can file an appeal, and shall have a high chance to succeed.
4) If the said house was appraised at $300k by 3 companies, yet she still like to make a $420k offer, so? Any law says appraised value has to be higher than an offer price?