Ali,
If you are on WordPress you can try some of these. First, set up the free Yoast SEO plugin if you have not already.
I have been working on this exact issue. The way Tags are set up (on the client site) we are getting duplicate pages. I am trying a couple of things:
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In Yoast XML Sitemaps > Taxonomies you can set "Tags" to "Not in Sitemap". Removing them from the XML Sitemap will at least lessen the frequency that Google accesses Taged pages.
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In Yoast Titles & Meta > Taxonomies you can select Index or Noindex for your Tags.
Following the Tag idea, you might get a 404 if a Tag is changed or deleted. It's hard to diagnose 404s without seeing them. Consider a custom 404 error page, if you have not already. That will at least improve the user experience. With htaccess you set different redirects and custome 404 pages depending on the site section.
Tags in WP don't seem to add much benefit, and probably cause more problems (like dupe content). Even if you are diligent with Tag consisitency, most blogs end up with nearly duplicate Tags.
There are many Moz posts on the topic of duplicate content in WP specific to Tags that might further illuminate your question.
Did you change anything universally, like switch to https? Missed redirects can cause a 404.
The WP Broken Links in Tools will direct you straight to the 404 pages.
Here are some more tips for dealing with 404 in WP:
http://www.wpbeginner.com/showcase/6-best-free-404-plugins-for-wordpress/
The WP plugin "Redirection" is well regarded, but use caution with lots of ecomm url variables.
Hope this helps!
