Why does the Boeing 737 have 5 x the passenger fatalities of the Airbus 320?
For one, the early 737s had a design flaw which allowed the rudder to reach a hard-over position, and get stuck there. Boeing initially dragged their feet in rectifying this, so it took a number of accidents before this design flaw was corrected. This is a major contributor to the high number of fatalities in the early operational years.

To have a more meaningful comparison, I would suggest to compare the fatalities per year, so the status in operational procedures can be excluded as a factor. As others have pointed out, this should give a more equal basis for a comparison.

If you look at fatalities per year, beginning in 2000, these numbers emerge:


Now you need to factor in the number of flight hours per type per year, which I do not have available. But I would expect that the low fatality rate in the early 2000s is more due to the small number of A320s around. But if we just compare the 737 NG with the A320, this advantage is reversed, and still the A320 comes out slightly ahead.

But thankfully, there are very few accidents for each type per year, so the statistical base is too low for a meaningful comparison. If we want to draw any conclusions, it could be that the A320 is indeed a safer aircraft, but the certainty of this conclusion is very low, being based on a very small number of accidents.