Relations is a feature that is built over another data using their geometry or metadata. Thus participants of relations can be points, ways, and other relations. They may present as physical objects of the world as well as mental objects, such as bus routes, borders, turn restrictions and etc.

So the way representing the road and tagged with highway=* tag can be a part of different route relations that run along it. Spliting or removing this way that is a member of relation will break all relations to which it belongs.

To avoid this you need to download all parent and child objects for this way (relation).

After you download all the data, JOSM helps you validate your edits and deal with possible conflicts.

Hit Update Modified (Ctrl+Alt+M) before any upload to avoid possible relation conflicts.

Fixing broken route relation

If you are faced with the broken relation of route and you have information about it, Relation Toolbox will help you cope with the work.

Select a route in Relation Toolbox dialog, then select missing part of the route and Press ➕ button to add it to the relation. After adding all missing segment open Relation Editor ✏️ to arange order of the route segments.

Press the button Sort the relation members on the left panel to re-order it. You need to rid of the gaps. If the route contains gaps, you need to find the missing parts. Also, you need to pay attention on the start and end points of the route (from=* & to=* tags ), if necessary, change the direction of the route by clicking on the button Reverse the order of the relation members.

If you do not have accurate information about the route passing you can consult with previous versions of the route. You can use for this purpose OSM History Viewer - http://osmhistory.appspot.com