It sounds like it may be an issue with a 3rd party plugin you have or customization.
I would check a couple of things:
Do you have any caching plugins installed?
If so, they need to be disabled during an update to ensure you are not being served outdated files once you make the update.
There may be a plugin conflict with a 3rd party plugin – you can check that by disabling then reactivating a plugin one at a time to see if you find a conflict there.
Also, it would be helpful if you could let me know what plugins and theme you are using on your website?