A reader writes:

Shape does not tell the whole story. It's tough drawing districts that fit all the guidelines: equal in size, respecting municipal boundaries, keeping together communities of interest, conforming to the Voting Rights Act, being compact and contiguous and all the other various criteria that are desirable and have little or nothing to do with predicting future voting patterns, etc. Sometimes an oddly shaped district is more diverse than a square one. It's content that counts.

Another writes:

It's not fair for you to highlight Arizona in this discussion

It is one of the few states that has a nonpartisan process for handling redistricting, which it has done effectively and without controversy.  For example, the district you show, AZ-2, is an historical anomaly; ever since Arizona got multiple congressional seats, the Navajo and Hopi nations - both of whom hate each other - have insisted on being in separate Congressional districts.  As Navajo territory completely surrounds Hopi, this is the only way of doing it.

Another:

Illinois' 4th district was the first majority Hispanic district in the state. The shape of the district was the result of a long, hard fight in which the Latino Community in Chicago sought to preserve their rights under the Voting Rights Act (King v. Illinois State Board of Elections). The district looks odd because it preserved the two largest Latino populations in Chicago, areas with Puerto Rican and Mexican-American majorities. The district successfully preserves a community of interest and does not dilute minority voting rights. Sometimes districts are oddly shaped for good reasons.

Another:

As you run through these peculiar districts, bear in mind that members of Congress represent people, not neat swaths of land. I don't mean to excuse egregiously manipulated districts, but might, for example, the California district that hugs the coast makes sense? It almost certainly is populated by people with a common interest related to the shoreline, and one likely muddled if you have too many inlanders included. Just a thought.

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