You can control this by creating menu items first for the Joomla articles category (blog/newsfeed or whatever), then for the tag component itself.
1. JOOMLA CONTENT
You want to have a page where the articles posted under the blog category will be displayed under the same URL scheme, like
For this you need to have a Category Blog menu item and set its category to the top category of your Blogs category section.
Now this category page will be accessible from the following url scheme:
Also, all articles of this category (unless they are linked directly with a menu item, or they belong to a category that is associated with another menu item that takes precedence over the one you just created check here for more info on this), will be accessible through the same scheme, disregarding from where they are linked:
Now to also control the where the tagged items will be accessible, you need the following:
A Tagged Items menu item (tags component), which you have to set it to display tagged items from all your tags.
If you want to display the listed tagged items, below the Blog menu item, then I would advise you to set the tags menu item as child of the your Blog menu item (set blog menu item as parent of the tags one).
Now when you click on one tag from your module, the Tags Items page will be displayed with the Blog articles items as parent.
Each article you click from this page, assuming it belongs to one of the blog categories, will be accessible following the above URL scheme, of the blog category, since it will be linked through that menu item:
However, here you may face a small issue, if you are still creating new tags while you are creating new articles, as the new tags won't be included in your tagged list of items.
One possible workaround is to define a standard list of tags that you want to use from within the tags component, select them all in your Tags menu item, and disable the AJAX creation of new tags from inside the com_content component, and use only tags from your list.
Otherwise, if you are creating new tags and those are not included in your tags menu item, most likely they will be displayed again under the home menu item - it won't affect where the actual article will be accessible though.
*The more advanced approach would be to create a custom menu item / override for the tags component, which will use all the tags by default, without the need to select them one by one.
**The above will take you closer to what you are after, I wrote them at once and in a hurry, so let me know if there is any non-sense anywhere or you need further explanations.