Fetching(GET) image url with applied image styles using jsonapi module
I would do it

create file: test_api.info.yml

name: Test API
type: module
description: Custom RESTful API module with GET/POST/PUT/DELETE etc.. method examples.
author: Minnur Yunusov
core: 8.x


create file: test_api.routing.yml

test_api.getparameter:
  path: 'my-api/get/{nid}'
  defaults: { _controller: '\Drupal\test_api\Controller\TestAPIController::getpara_example' }
  methods:  [GET]
  requirements:
    _access: 'TRUE'   


create file: /src/Controller/TestAPIController.php 

&lt;?php

/**
 * @file
 * Contains \Drupal\test_api\Controller\TestAPIController.
 */

namespace Drupal\test_api\Controller;

use Drupal\Core\Controller\ControllerBase;

use Symfony\Component\HttpFoundation\Request;
use Symfony\Component\HttpFoundation\JsonResponse;
use Drupal\user\Entity\User;

/**
 * Controller routines for test_api routes.
 */
class TestAPIController extends ControllerBase {



  /**
   * Callback for `my-api/get.json` API method.
   */
  public function getpara_example($nid, Request $request ) {

    //get node;
    $node=node_load($nid);

    //get image uri
    $image =$node-&gt;field_image-&gt;entity-&gt;getFileUri();

    $style = \Drupal::entityTypeManager()-&gt;getStorage('image_style')-&gt;load('thumbnail');

    $url = $style-&gt;buildUrl($image);


    $response['nid'] = $nid;
    $response['uri'] = file_create_url($image);
    $response['image_original'] = file_create_url($image);
    $response['image_style_thumbnail'] =file_create_url($url);


    return new JsonResponse( $response );
  }




}


test:
url: localhost/drupal8/my-api/get/30
method: get

note: 30 is nid.

result:

{
  "nid": "30",
  "uri": "public://2017-10/generateImage_1RwT72.png",
  "image_original": "http://jeremiselxi.260mb.net/drupal8/sites/default/files/2017-10/generateImage_1RwT72.png",
  "image_style_thumbnail": "http://jeremiselxi.260mb.net/drupal8/sites/default/files/styles/thumbnail/public/2017-10/generateImage_1RwT72.png?itok=Dzg02oew"
}