Here is a website which has instructions on how to run a variety of web browsers in private mode:

https://www.howtogeek.com/269265/how-to-enable-private-browsing-on-any-web-browser/

Google Chrome:

To open an Incognito Window in Chrome, open the Chrome Menu by clicking on the 3 dots in the upper-right-hand corner of the browser window and select New Incognito Window. You can also use the shortcut Shift + ⌘ + N (on macOS), or Shift + CTRL + N (on Windows/Linux).

https://support.google.com/chrome/answer/95464?hl=en&co=GENIE.Platform%3DDesktop

This message will appear after you select New Incognito Window in Chrome:

Microsoft Edge:

In Edge, this is called InPrivate window.  To open an InPrivate window in Edge:

1. Open the Edge Menu in the upper-right-hand corner of the browser window by clicking on the 3 dots.
2. Select New InPrivate Window.

https://support.microsoft.com/en-us/microsoft-edge/browse-inprivate-in-microsoft-edge-cd2c9a48-0bc4-b98e-5e46-ac40c84e27e2