Browser

it is not the browser that is so stating. It is coming from web sites. Please allow me to explain. It is important to know that Safari is installed as part of the operating system and it normally it cannot be individually upgraded. It is periodically updated when the computer's operating system (OS) is updated or upgraded. If you are unable to access certain web sites  and getting a message that your browser must be updated, this is not an unusual situation. The reason is that web site engineers write their software to work with specific browsers and specific versions of them. You can hope that your web site's developers eventually update their site to work with Safari, or you can use another browser that does work such as Firefox or Chrome. They can be individually updated as needed. Finally, even if you are able to update Safari, there is no guarantee that it will be compatible with every web site. I hope that this is helpful.

Which browser, if you are running High Sierra the browser you are using may not meet the security protocols required by the webpages you are visiting, that could be the source of the error message.