We sympathise with your problem...
Often the safest and most comfortable cycle routes in London follow back streets, avoiding main roads. However, it's difficult to know a city as large as London well enough to ride in back streets all the time, and these are sometimes too slow for longer journeys.
For these reasons, London cyclists are often likely to find themselves following main roads, because these are fast and are easy to navigate. However, there's clearly an increased risk of a collision on main roads and at major junctions. This is a key reason why Dutch-style segregation on main roads and at major junctions is necessary, because these roads will always be popular cycle routes.
A smartphone with maps and route-planning is a useful tool to help find less busy routes that might also be direct. A web browser
As mentioned above, apps like Cyclestreets and Bikehub (they use the same mapping algorithms) can provide a choice of routes that have effectively been chosen by other cyclists.
Indeed, we use the same route-planner on our home page, where you can plan cycle-friendly routes from anywhere to anywhere.
Another solution is to use Transport for London paper maps (which were originally created using data from our activists). These routes are far from perfectly connected, but they do often provide a good compromise between speed and comfort, using what cycling-specific facilities exist (such as the incomplete but still useful LCN+ routes).
In practice, there's likely to be significant crossover between the Cyclestreets route choices and those on the paper maps.
To sum up: putting extra effort into route-planning is likely to provide safer and more confortable routes, although sometimes at the expense of convenience and directness.