Each time I run the command "docker-compose run web ..." it results in another container being added to Kitematic, so that I have a list of containers like "image-name_web_run_1" and so on with each run of docker-compose.
Is this expected behaviour? If so, how do I work around it?
It is expected. You can use docker-compose run --rm to have the container removed when it exits.
You can clean up these old containers using docker-compose down or docker-compose rm -a.