Firefox allows you to automatically enable private browsing mode via its options window. Click menu > Options to open it.
Click the “Privacy” tab at the left side of the window to access privacy settings. Under History, click the “Firefox will” box and select “Never remember history”. Firefox will prompt you to restart the browser.
Firefox will look the same but it will have the same settings it uses in private browsing mode applied.
Google Chrome
To activate Google Chrome’s incognito mode by default, you must add a command line option to its shortcut.
First, locate the shortcut you use to launch Google Chrome—either on your taskbar, desktop, Start menu. Right-click it and select “Properties”.
Add -incognito to the end of the text in the Target box. That’s a space, one dash, and then the word incognito.
Click “OK” to save your changes after adding this option.
Google Chrome will now start in incognito mode when you launch it from this shortcut. If you use other shortcuts to launch Google Chrome, you will also need to modify them.
Apple Safari
The Safari browser on Mac includes an option that allows you to always open it in private browsing mode. To find it, open Safari and click Safari > Preferences.
On the General pane, click the “Safari opens with” box and select “A new private window”. When you open Safari in the future, it will open in private browsing mode.
Internet Explorer
If you’re using Internet Explorer, you will need to add a command-line option to your Internet Explorer shortcuts to activate InPrivate Browsing by default.
Locate the shortcut you use to launch Internet Explorer, right-click it, and select Properties. If you’re using a taskbar shortcut, you’ll need to right-click Internet Explorer on the taskbar, right-click “Internet Explorer” again, and select Properties.
[reference: https://www.howtogeek.com/137466/how-to-always-start-any-browser-in-private-browsing-mode/ ]
