The appraised value from the Appraisal District is done by a method known as "mass appraisal" where the property is given a value based on the non-distressed sales in the nearby area. Mass appraisals are done from the outside -- no inside evaluation of condition or amenities.
Since foreclosure sales are ignored for mass appraisals, the market value of a house might be a little high if there are distressed sales in the area.
You have used the right word "normal" to characterize prices. They are a distribution about some median price, some higher, some lower. The time of year affects the median as well as condition and features of a property. A common myth is that market value is a single number. A sales price will fall within a range of numbers, and 'the' value on an appraisal should be considered as having a "plus or minue" after it.
Negotiating is a whole other art from estimating value. Knowing what your maximum price to pay is a good place to start. Discuss with a qualified Realtor what the house should sell for next. If your price is below the should-sell price, ask if negotiation can result in your price or not.
There is no right or wrong. The seller might take your offer. If you can understand the seller's position, such as an heir who has no ties to the property and no mortgage to pay, or, say, a hard-pressed couple who just bought the house last year for what they're asking, you can guess what the answer will be about taking $10-20k off.
Your Realtor probably has the tools to advise you and negotiate for you.