I am using Firefox web browser most of the time, but you need to know that I have drawn a clear line between daily browsing and strictly private usage by using browser sandbox. I will shortly describe whole process so you could also benefit from it.

Step 1

Create browser_sandbox user which will be used to execute iceweasel browser.

$ sudo useradd -m browser_sandbox

Step 2

Install Xephyr (X server outputting to a window on a pre-existing X display).

$ sudo apt-get install xserver-xephyr

Step 3

Modify sudoers file to run commands as a browser_sandbox user without a password.

$ sudo visudo

milosz ALL = NOPASSWD: /bin/su - browser_sandbox
Replace milosz user with your username.

Step 4

Adapt and use the following shell script to start icewasel in the sandbox mode.

#!/bin/sh
# Browser sandbox
# Do not forget to edit sudoers file
# Description can be found at https://sleeplessbeastie.eu/2013/07/19/how-to-create-browser-sandbox/
# display used to show browser
SANDBOX_DISPLAY=":33"
# sandbox user
SANDBOX_USER="browser_sandbox"
# authentication cookie
SANDBOX_MCOOKIE=`mcookie`
# authentication file
SANDBOX_XAUTH="/home/milosz/.Xauthority-sandbox"
# local X11 sockets, used to determine if application is running
LOCAL_SOCKETS="/tmp/.X11-unix"
# window width and height
WIDTH=1000
HEIGHT=700
# application to start
# width and height are raised above window dimensions to occupy full window
APPLICATION="iceweasel -width $(expr $WIDTH + 1) -height $(expr $HEIGHT + 1)"
# clear authentication files
clear_auth_files() {
  # clear authentication file
  if [ -e $SANDBOX_XAUTH ]; then
    unlink $SANDBOX_XAUTH
  fi
  # create empty file
  touch $SANDBOX_XAUTH
  # clear authentication file for SANDBOX_USER
  sudo su - $SANDBOX_USER << EOC
    if [ -e .Xauthority ]; then
      unlink .Xauthority
    fi
    touch .Xauthority
EOC
}
if [ ! -e ${LOCAL_SOCKETS}/X${SANDBOX_DISPLAY#:} ]; then
  # display is not active
  # clear stalled authentication files
  clear_auth_files
  # store authentication cookie for chosen display
  xauth -f ${SANDBOX_XAUTH} add ${SANDBOX_DISPLAY} . ${SANDBOX_MCOOKIE}
  sudo su - $SANDBOX_USER << EOC
    xauth add ${SANDBOX_DISPLAY} . ${SANDBOX_MCOOKIE}
EOC
  # start Xephyr and application
  Xephyr -auth ${SANDBOX_XAUTH} -screen ${WIDTH}x${HEIGHT} -br -nolisten tcp $SANDBOX_DISPLAY &
  sudo su - $SANDBOX_USER << EOC
    DISPLAY=$SANDBOX_DISPLAY $APPLICATION
EOC
  # clear authentication files after session is closed
  # do not start application in the background to use this function
  clear_auth_files
else
  # display is active so application is already running
  # show simple error message
  if [ -n "$(which kdialog)" ]; then
    kdialog --error "Application is already running.<br/>Display ${SANDBOX_DISPLAY#:} is active."
  fi
fi

The idea is quite simple as you need to start Xephyr in the background and then run iceweasel as browser_sandbox user on the created display.

You can extend it to clear or reset browser settings, use memory-based file system or start different application.

Clipboard

You need to install xclip utility to read or modify clipboard in the sandbox:

$ sudo apt-get install xclip

To read sandbox clipboard (display :33) execute command:

$ XAUTHORITY=~/.Xauthority-sandbox xclip -selection clipboard -o -d :33

To set sandbox clipboard execute command:

$ echo "http://www.debian.org" | XAUTHORITY=~/.Xauthority-sandbox xclip -selection clipboard -i -d :33