Your issue is a combination of insulation and venting. The ice dam is
caused by your attic air being warmer than the outside air in winter.
The warmer air melts the snow which runs toward your eaves which of
course are surrounded by outside air. The snow melt hits the colder
eave and freezes. In a short time this builds up and holds water back
behind it. As the typical shingled Minnesota roof is not designed to
hold water - the water leaks into the house.

Now, I don't know if you need more insulation or more venting - but the
goal is to keep the attic air the same temp as the outside air in the
winter.