The macOS version you can upgrade to depends on the year your Mac was released.
Click the Apple menu () at the top of your screen and choose About This Mac.
The 2013 MacBook Pro cannot be upgraded beyond macOS 11 Big Sur. And you may upgrade to 10.14 Mojave or 10.15 Catalina as well.
➡️ macOS 10.14 Mojave is compatible with the Macs listed here
➡️ macOS 10.15 Catalina is compatible with the Macs listed here
➡️ macOS 11 Big Sur is compatible with the Macs listed here
➡️ macOS 12 Monterey is compatible with the Macs listed here
➡️ macOS 13 Ventura is compatible with the Macs listed here
➡️ macOS 14 Sonoma is compatible with the Macs listed here
❗️ – IMPORTANT – None of your 32-bit apps will work in macOS Catalina or newer. This includes older MS Office apps and any other older software apps and drivers. These will require that you find 64-bit updates and/or replacement titles.
32-bit app compatibility with macOS High Sierra 10.13.4 and later - Apple Support
Use the free utility Go64 to check the 64-bit compatibility of your apps:
https://www.stclairsoft.com/Go64/
❗️ Do NOT upgrade
…until you have made a Time Machine or other backup of your startup drive.
Learn How to Back up your Mac with Time Machine
✅ - Download the appropriate macOS upgrade installer app:
Click > 10.13 High Sierra, 10.14 Mojave, 10.15 Catalina, macOS 11 Big Sur.
Once downloaded, double-click the “Install macOS…” app in the Applications folder to begin the upgrade.
Please see > How to download macOS - Apple Support.
Also, if you are having trouble connecting to and using certain websites with your older version of Safari, try it with a newer third party browser that is still being updated to meet current internet security protocols.
I suggest you take a look at Firefox browser or Brave browser.
Download an alternative browser and install it into your Applications folder. Launch and use that when you encounter websites that don't play well with Safari.