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To upgrade to latest Safari, you need to upgrade the current OS X to latest version. The latest OS X is 10.11 El Capitan. Upgrading to El Capitan also update Safari and other browsers.
Follow the steps below to upgrade the OS X if your Mac support it.
Click on the Apple icon and App Store. This will bring App Store. Click Updates tab. If your Mac can handle latest OS X, then you will see Mac OS X El Capitan there with FREE UPGRADE button. Click Free Upgrade and enter your Apple ID and password.
This will download the update. It may take a few hours to download. Once the download is completed, it will bring the OS X Installer. Follow the onscreen instructions to install it. It may take up to 3 hours to install the upgrade.
If you have any difficulty on following the steps above, please let me know and I will continue to help you.
No problem. Take your time and leave it turned on. To make sure the Mac doesn't go to Sleep mode, go to System Preferences > Energy Saver and set Sleep to Never. You can change it after upgrading.
If you are able to start the download, that means your Mac does support latest OS X. So the download will definitely work. After the download, it may take up to 2 hours to install it.
Make sure you don't turn off your Mac or internet connection. It need a stable internet connection to complete the download or you will have to start over. I see that you are not able to run videos on web like Youtube. I've a direct fix for it and you can apply it now.
We can uninstall old flash player and install a new version.
Click this link and this will download the flash player uninstaller file --> https://fpdownload.macromedia.com/get/flashplayer/current/support/uninstall_flash_player_osx.dmg
After downloading, click on the Finder icon on dock, then click on Go menu and select Downloads. This will bring Downloads folder. Now you see a file named "uninstall_flash_player_osx.dmg" there. Double click on it to open. Then open flash player uninstaller file. Let it uninstall
Then download latest flash player from this link --> https://fpdownload.adobe.com/get/flashplayer/pdc/126.96.36.199/install_flash_player_osx.dmg
After downloading, click on the Finder icon on dock, then click on Go menu and select Downloads. This will bring Downloads folder. Now you see a file named "install_flash_player_osx.dmg" there. Double click on it to open. Then open flash player installer file and install it. Then you will be able to watch Youtube videos.