Permalink not changing according to post type

Permalink structure for blog posts and pages are managed in WordPress Settings -> Permalinks page, so to make blog posts have permalinks like blog/%postname%/ and you’ve done it right.

The tricky part is about Custom Post type permalinks declaration:

in your CPT arguments declaration there’s a rewrite rule parameters, which in turn has with_front – if this parameter is false you’ll have permalink structure without prefix used for blog posts, and if it’s true you’ll have what you’ve experienced. By default it’s set to true.

So fixed part of rewrite parameter will look like.

'rewrite' => array(
    'slug'       => 'casestudies', // your case studies permalink base.
    'with_front' => false, // important to make your links without base of blog posts.
    'pages'      => true, // set true if you need pagination for your cpt listing.
),