What you call "normalized," I would call "over-normalized." In none of the examples in the document you reference do they suggest having a field name be a column value. Each column should have values that would be considered together in a set to be aggregated in some way (sum, average, etc.). Temperature and ice cream sales may be correlated in some way, but you would never add a temperature to ice cream sales. Every value in a measure column should be in the same units (e.g. currency in dollars, weight in pounds, quantity, etc.). Dimension values should all be values in the same category (e.g. names of cities, flavors of ice cream, etc.).
A more common problem encountered by new Tableau users is to have multiple columns for values that should be a single column, for example they have separate columns for sales in 2010, 2011, 2012, 2013, 2014, etc. This frequently needs to be reshaped or "pivoted" to have a column for "year," and a column for "sales." This is what the old Excel reshaper tool did, and what the new data connector does in 9.0.
I hope that helps.
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Depending on what you're trying to do, you can achieve this without transforming your data by simply using Field on a number of shelves. E.g. you can show average ice-cream sales and average temperature on the same chart as different color line by dragging Field onto Color shelf.
If you want to use different aggregations or use them in the same sheet in different calculations you can break them up by writing a calculation like
IF [Field]='Temperature' then [Value] END
And another calculated field like
IF [Field]='Ice Cream Sales' then [Value] END
This way you're breaking each field into a separate column since each of these fields only return the values in the [Value] column if field is of particular type.
If you really want to transform your data, you can use Custom SQL option in the Data tab and write a SQL query that takes advantage of PIVOT function on SQL Server.