for safari, should the web browser be updated automatically with any iOS update?

The Safari browser sometimes gets its own update, but typically is updated with a macOS update or upgrade for Mac computers. Of course, that only happens if your Mac can support the newer macOS version. But you may not actually have to upgrade the macOS to get your website access back.

If you are having trouble connecting to and using 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.

Note - iOS is for the iPhone or iPad, not the Mac.

Chekeung wrote:

my bank ask me to update my browser.

The bank's website is defective. Contact them and tell them to fix it.

Update to the latest version of Safari - Apple Support says,

If a website says that Safari is out of date even though you're already using the latest version of macOS or the latest version of iOS or iPadOS, there could be an issue with the website. If you’re sure that you want to use the website, contact the website owner or developer.