What is a free site where I can upload shapefiles and make my own interactive map to share with others by a link?
You can create a free public account on ArcGIS.com. From there, you can upload your shapefile(s) (provided that they contain fewer than 1000 features), create maps containing your shapefiles, and share your maps with others via URL.
If you can live with some styling issues, you can save the shapefiles as KML and upload them to MyMaps on Google Maps.
If you are into open source, you can also use CartoDB, which allows you to import shapefiles as well as use other GIS formats. Quoting wikipedia:
CARTO (formerly CartoDB) is a Software as a Service (SaaS) cloud computing platform that provides GIS and web mapping tools for display in a web browser
I've been worked with ArcGIS Online having a free public account. There you can upload the shapefiles and then you can share the link with your customers, friends, colleagues etc.