On the website, go to the "my" tab, select "journeys".

That should show a list of all your journeys (and folders if you use them). 

On the extreme right is the wastebasket symbol. Click that to delete. 

Still didn't work.

I am logged in on the web version and can see all my saved routes bottom left of the map.  (they are also all available on the phone).  But if I click on My Bike it says "no such user".

In that case, have you tried the old "switch off and start up again" method of problem solving? :-) Or log out of CT then back in. 

This one’s a mistake on my part, I’m afraid! Accounts created with the Android app weren’t properly registering with the editing pages on the site. That’s fixed now so you should be able to use the method Hobbes describes to delete a route.

(In a future version of the app you'll be able to delete routes directly by swiping left in the list.)

That all worked.  Thanks.