Firefox has menu items that use the Shift and Ctrl modifier keys (e.g. Ctrl+Shift+Z = Redo) and those can interfere, especially when typing in a text area. You might have more luck with Alt and Shift instead.

Try to assign different keyboard shortcuts to each of the installed languages instead of using a key sequence to cycle between installed language layouts.

Also Alt+Shift doesn't help. All was working fine for years until 2 days ago. I haven't installed any new program recently. Also it all works fine in Word and other applications. The problem is only when typing in browsers. Different key sequences do work, but why Alt+Shift has stopped working suddenly in browsers ??

Ctrl+Shift is used for some menus and Alt+Shift is used for access keys on web pages (some forums use for instance the 's' access key to submit a reply: Alt+Shift+S).