Add this to your functions file:
function remove_json_api () {
// Remove the REST API lines from the HTML Header
remove_action( 'wp_head', 'rest_output_link_wp_head', 10 );
remove_action( 'wp_head', 'wp_oembed_add_discovery_links', 10 );
// Remove the REST API endpoint.
remove_action( 'rest_api_init', 'wp_oembed_register_route' );
// Turn off oEmbed auto discovery.
add_filter( 'embed_oembed_discover', '__return_false' );
// Don't filter oEmbed results.
remove_filter( 'oembed_dataparse', 'wp_filter_oembed_result', 10 );
// Remove oEmbed discovery links.
remove_action( 'wp_head', 'wp_oembed_add_discovery_links' );
// Remove oEmbed-specific JavaScript from the front-end and back-end.
remove_action( 'wp_head', 'wp_oembed_add_host_js' );
// Remove all embeds rewrite rules.
add_filter( 'rewrite_rules_array', 'disable_embeds_rewrites' );
}
add_action( 'after_setup_theme', 'remove_json_api' );