Okay, after having a better overview of you issue, here is what you need to do to achieve what you want:
Create a template override inside the html folder of your template, for the Tags Component.
To do this, create the following folder structure:
/templates/yourtemplate/html/com_tags/tag/
Copy in the above folder the file:
default_items.php
from the components/com_tags/views/tag/tmpl folder.
Open this file to edit it. Around the line 80 you will see:
<?php if ($this->params->get('tag_list_show_item_description', 1)) : ?>
<?php echo $item->event->beforeDisplayContent; ?>
<span class="tag-body">
<?php echo JHtml::_('string.truncate', $item->core_body, $this->params->get('tag_list_item_maximum_characters')); ?>
</span>
<?php echo $item->event->afterDisplayContent; ?>
Change it by adding your code before echoing the description text (e.g. before the "tag-body" span):
<?php
$item->core_body =strip_tags( $item->core_body, '<img>');
?>
So you will end with something like below:
<?php if ($this->params->get('tag_list_show_item_description', 1)) : ?>
<?php echo $item->event->beforeDisplayContent; ?>
<?php
$item->core_body =strip_tags( $item->core_body, '<img>');
?>
<span class="tag-body">
<?php echo JHtml::_('string.truncate', $item->core_body, $this->params->get('tag_list_item_maximum_characters')); ?>
</span>
<?php echo $item->event->afterDisplayContent; ?>
Initial Answer:
You website is a bit confusing, a bit hard to navigate and there is not a standard way to reach to all kind of content and categories. I also think that there is a bit of confusion in the configuration, as some pages are displaying the tags info, other only the categories and so on.
Furthermore, I was trying to reach one of the articles from that tag list to examine how and why it uses an image in its introtext with link to external websites, but although the images of the articles direct to external websites, the article titles produce a 404 error.
First of all, I believe you should refine the whole configuration of the website and the components. I can't say what exactly you need to do within this answer - you probably might need to hire someone to help you sort this out.
Regarding the issue with the links - I checked also a few other tags and not all tags are having links into their description. This leads me to think that those specific images into the introtext are linking to these external websites.
I think that since you have set the Tags component to display the introtext, it will just use that and will contain any of the markup of the article's introtext.
One way to overcome this, is to use the hide introtext feature of the Content component.
Following this approach, you will be creating the same content twice for introtext and fulltext. In the introtext you will be using the image without the link, while in the fulltext you will be able to add your link. Setting Joomla to hide the introtext from the article display, will result to have 2 different block of texts to display for your needs.
The introtext without the links for all blog layouts/ tag lists, article modules etc and the fulltext for the article display.
Or you could use the Joomla Content's images features, from the tab "Images and Links" and set there your intro image.
However keep in mind that the output of each component is also controlled by any possible template overrides for each component.