Find if there is any compatibility issues, and resolve them prior UI upgrading
Actual UI Upgrade
Upgrade-SPSite http://SPWeb.Com/sites/siteCollection –VersionUpgrade -Unthrottled
#Unthrottled flag starts upgrading immediately
Find the current upgrade status
Get-SPSiteUpgradeSessionInfo –site http://...
Have you also updated the site collection? https://docs.microsoft.com/en-us/sharepoint/upgrade-and-update/upgrade-a-site-collection-to-sharepoint-2013
Once you upgrade from SharePoint 2010, the sites will work in SharePoint 2013, but the look and feel won't be updated until you Upgrade the site collection. You can also do this through Powershell:
SharePoint Modern Sites requires "SharePoint Framework(spfx) Extensions" in order to extend SharePoint User Experience. It's the modern SharePoint alternative to "Master pages".
You can find official Microsoft Documentation on extensions here.