Can't find out base GitLab API base url

The correct base url for the hosted GitLab is https://gitlab.com/api/v4/ so your request to
GET /projects would be

curl --header "PRIVATE-TOKEN: XXXXXX" "https://gitlab.com/api/v4/projects"

That would return all projects that are visible to you, including other user's public projects.

If you wish to view just your projects, then you should use the GET /users/:user_id/projects endpoint, where :user_id is your user ID that can be found on your GitLab profile page or in the response to your request to GET /user if you're authenticated.

# Get :user_id from this request
curl --header "PRIVATE-TOKEN: XXXXXX" "https://gitlab.com/api/v4/user"

# See your projects by replacing :user_id with id value from previous request
curl --header "PRIVATE-TOKEN: XXXXXX" "https://gitlab.com/api/v4/users/:user_id/projects"

Also, the project ID is not the same as the project name. You can retrieve the project ID from the response of your request to GET /users/:user_id/projects, or from the project's settings page.

For me the following request worked:

curl --header "PRIVATE-TOKEN: YOUR_TOKEN" "https://gitlab.com/api/v4/users/YOUR_USER_ID/projects" 

Don't know why the request :curl --header "PRIVATE-TOKEN: PRIVATE_TOKEN" "https://gitlab.com/api/v4/projects/" returned a list with some other public projects.

Another useful request for user info: curl --header "PRIVATE-TOKEN: PRIVATE_TOKEN" "https://gitlab.com/api/v4/user/"