uMap lets you create maps with OpenStreetMap layers in a minute and embed them in your site.
uMap overpass integration
There is a session about uMap in the afternoon of the conference: Nutzung von OpenStreetMap für Standortkarten und Online-Stories
Go to https://umap.osm.ch to play around with umap.
Hack night project
It is possible to include OSM data from overpass as a dynamic layer. This hack night project aims to include a simple interface to create such a layer for a key=value overpass search query. See issue 112