Hi,
Usually private browsing mode isn’t something you select. It sometimes useful in helping determining login issues. When you browse the Internet in any browser, your browser stores information about places you’ve visited, etc. Sometimes that can cause login issues depending on a number of circumstances. If you’ve gone into private browsing mode and see that the issue is now gone one thing you can try to do is clear the cache in your normal browsing mode.
Here is an article that details that:
https://www.imore.com/how-clear-stored-website-data-ios-7-safari
I would caution you in that clearing cache can wipe out other information such as stored passwords. However, if you’re not using mobile Safari other than for casual browsing this may not be a huge issue for you. Please keep in mind that I’m providing this information based off of other login issues I’ve had with other sites. As such your mileage can vary. If you really want to do another acid test I would recommend going to the App store and downloading the Google Chrome browser and try logging in from there. This will be an extra layer of confirmation that clearing the cache may be a solid option.
Hope all is well!