Approximately 7 years ago the Link Manager option was dropped from WordPress core. This was replaced by a Link Manager plugin. This plugin has not been updated since WordPress 3.5 and would appear to be no longer supported. The latest version of WordPress (version 5.1) appears to have lost the Link Manager from the menu system and the Links widget from the widget dashboard. This means that links that have been added to blogs will no longer be visible.
In order to display a list of links in a blog you can create a custom menu in Appearance > Menus and then display the menu in a Navigation Menu widget.
Update Wednesday 6th March 2019
The WPMU Dev Branda plugin (formerly Ultimate Branding) includes a Link Manager plugin. Go to Network Admin > Branda > Admin Area and set the option to Enable Link Manager. This will restore Link Manager to all blogs and the links widget to Appearance > Widgets. Furthermore, it retains all links that you may have previously created. All you have to do is then go to Appearance > Widgets and drag and drop the Links widget into the sidebar where you wish it to appear.