When I create a new web map (Content>My Content>Create>Map) Portal automatically adds a feature layer and a topographic layer from my data. I cannot delete either of the layers. Is there a default map template or something that I can edit or delete to have this behavior stop?
ArcGIS Server 10.6.1
That sounds like the base map, which is chosen by your portal's administrator.
If your admin made multiple base maps, you can switch between them:
You can also switch it off: