Click on the image will jump to another site, but the site will be retained, like https://www.voucherspromo.co.uk/discounts/wilko.com, many sites have this function.
You could add the target attribute in the anchor tag with the value _blank.
<a target="_blank" href="http://your_url_here.html">Link</a>
This will open all your links in new tab and retains the current page.
It is worth noting that you can also achieve a similar (and perhaps smoother) effect with frameworks like React for simple apps using client-slide routing, where most of the resources are downloaded (if I’m not mistaken) and cached on page load. For more complex apps and websites you would still fetch data externally from an API, which isn’t too different from what was mentioned above.
Yes you are right clicking on the image will jump to another site, like https://www.freedeliverydeal.co.uk/discount/ax-paris/