Hey, I found the street, which is after the search highglighted just in its part. How can I add the highlight for all its length?
Streets in OSM can be divided into many lines. The reason is e.g. different surface (or other properties like speed limit) on different parts.
You can do what you need with Overpass Turbo.
See: http://overpass-turbo.eu/s/Egv (click Run)
On the top is a query which you need to enter into the query wizard - in this example to search for street named Starowiejska in the current map view.