Step 1 - Go to settings

Hover over the small thumbnail image of your profile picture and then click on 'review' to the right of Privacy & Settings

Step 2 - Enter your password

Step 3 - Click on the 4th setting under Profile > Privacy Controls (Select who can see your connections)

This is the screen you automatically see;

Step 4 - Change the setting to 'Only you'

Then click 'Save changes'

Step 5 - Ask a connection to check whether they can see your connections

They need to go to your profile and click on where it shows how many connections you have;

Step 5 - Ask a connection to check whether they can see your connections

They should only see shared connections (if there are any)

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Step 6 - Clear cookies

If your (not shared) connections are still showing you may want to both log out of Linkedin, clear your browser cookies and history and then log back in.