Why is DNS a privacy concern?

The DNS is one of the most significant leaks of data about an individuals activity on the Internet.

Some of the issues in simple terms:

An overview of the problems is given in this Tutorial: DNS Privacy Tutorial.

For an expert review of this topic recommended reading is DNS Privacy Considerations.

The solution?

For a full discussion of the options available please see DNS Privacy - The Solutions.

Client (stub) to recursive resolver

The two most widely deployed solutions for stub to recursive resolution are DNS-over-TLS and DNS-over-HTTP; they both encrypt DNS data and prevent passive surveillance of network data revealing users' DNS queries. They can both allow users to validate the server they choose for their DNS service to make sure they are using a provider who has a good privacy policy for how they handle user data. But they do have some different protocol properties and in practice are being deployed in somewhat different ways at the moment. Neither of these are trivial changes in the way DNS works and encryption of all DNS queries by default will not happen overnight. 

See DNS Privacy ClientsDNS Privacy Implementation Status,  DNS Privacy Public ResolversDNS Privacy Test Servers for more information.

Recursive resolver to Authoritative server

The DPRIVE working group at the IETF has been working on solutions for that, if you are interested see the DPRIVE mailing list. 

SNI

Unfortunately the Server Name Indicator header in HTTPS messages also reveals the name of the website contacted by the user so provides a similar leakage channel for web traffic as the DNS queries. However there is work underway in the TLS working group at IETF to encrypt the SNI: I-D: Encrypted Server Name Indication for TLS 1.3.