What's your definition of a "reasonable" amount of data. 🙂
The data you've uploaded creates a 6MB web map, which is big (for web browsers). This data has to be downloaded and drawn on the browser. This won't work at all in IE as you've experienced because it's more than IE can handle. Chrome and Safari will be fastest, Firefox next.
The short term solution is to generalize your data before uploading it (you can do this with ArcGIS Desktop). Even though your data may not look like a lot, it really depends on the amount of vertices in the lines. Generalizing the data could significantly reduce the size.
By next week, we will have released an update to arcgis.com that will allow you to generalize the data in the web app. You'll see an option on the "Add data from file" dialog box. (Also, you won't be able to add this same shapefile next week because of its size, as we'll be setting a size limit for exactly these performance reasons you've encountered.)
Can you share your shapefile with us for testing purposes?
Thanks,
Mike
