How can I set the current working directory for docker exec with an internal bash shell?

From API 1.35+ you can use the -w (--workdir) option:

docker exec -w /my/dir container_name command

https://docs.docker.com/engine/reference/commandline/exec/

You can achieve it with:
docker exec -it containerName sh -c "cd /var/www && /bin/bash"

When building an image, one can specify the WORKDIR variable in Dockerfile:

WORKDIR /var/www/html

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