Bicycle routes in Europe

The data quality of OpenStreetMap (and other map services) varies a lot from country to country. Particular problems I have run into with OSM when bicycling are:

• In many European countries, bicycling is often prohibited along better or high traffic roads. OSM maps usually have no data on this, making it impossible to know if you are allowed to ride along a specific road.
• Bicycle paths on OSM maps are of unpredictable quality. I have often run into bicycle paths, for which you probably would need a mountain bike and which could at least not be driven with a loaded road or trekking bike.
• A really odd thing with OSM is that at least a few times, I had planned to use roads that simply do not exist. I am not sure how they ended up on the map.

Google Maps is slightly better, at least for some countries. If you use the route planner for bicycling, Google Maps will avoid roads where bicycling is prohibited.

Depending on exactly where you are going and if you are planning to follow designated long-distance cycle routes like the EuroVelo network or national cycle route networks, you can often find designated web sites for these cycle routes with much more relevant and up to date information than on any free online map. I am going through France next month by bicycle and one example of such a web site would be 'Loire by bike'. If you speak German, I can also recommend radreise-wiki.de with lots of general information about countries and regions and also detailed first-hand information on many cycle routes.

I would go for a bike route planner for the big distances and go with the bike routes they suggest or the maps, for the actual routes you use.
The European Bikeroutes planner information page, (much in Dutch but also in English and German) or a link to the planner (which has a tri-lingual interface with just a bit more Dutch) combines long distance routes and suggests, your choice between them, signposted routes, written routes in a selection of languages, GPS routes and possibly others as well. And these days they also have an app for your phone.

For planning details I do use a local bike route planner as well as Google (or open street map.)
Out on the street I prefer to have a local 1:50,000 map, by preference one that is made for cycling use.

I'm using outdooractive.com quite frequently. It has bike paths incoorporated and tries to route you along them where possible. It requires to open an account with them, which is free of charge.

E.g. for a trip from Gent to Budapest this would look like this

and of course you may zoom in as much as you like and set custom way points, export the GPX track etc.

For detailed regional planning I also use opentopomap.org, which has hillshading and isolines, very useful when planning bike trips. There is no routing available, but you may load GPX tracks into the map.

(from such map it may be more obvious that it's better to take the detour along the river than the shortest track. ;-))

If you're travelling through the Netherlands Fietsnetwerk is a great option. They have a nice app that works well with cycling touristic routes.