Looking into the effect of government party on recent Covid19 cases. Starting with the simple stuff.

Seven day average of new cases (as of today) per 100k people:

GOP governor: 9.9
Democratic governor: 6.1

That's about 60% more cases/capita in states run by a GOP governor.
Same as above but with party control over the legislature.

Seven day average of new cases (as of today) per 100k people:

GOP legislature = 10.1
Democratic legislature = 5.4

(it's 4.6 in the two states with a split legislature)
What happens if the same party controls the executive and legislative branches? Here things get more interesting.

GOP controls both = 11.5
Democrats control both = 5.9
Neither party controls both = 5.2
Next is comparing GOP control of different branches of the state government.

GOP controls executive and legislature = 11.5
GOP controls executive, but not legislature = 3.2
GOP controls legislature, but not executive = 6.3
A lot of people claim that the reason GOP-controlled states have more Covid19 cases today is because they had fewer early cases on. That may well be, but there's actually a very weak relationship between total cases on May 1 and number of new cases in the past week.
Here's a basic regression looking at the effect of GOP control over a state's executive and legislature branches on the number of new cases/capita. Note the effect of population density and the number of cases/capita from May 1 (a proxy for large-scale early exposure).
Controlling for population density and early exposure to the virus, having full GOP control over state government is associated with between 2.3 and 9.6 more Covid19 cases/day/100k people. I.e., a GOP-run state with 1 million people can expect from 230 to 960 more daily cases.
Not really sure what to make of this, but warmer states have more new Covid19 cases per capita than cooler states. This holds for states controlled by the GOP (13.8 vs. 6.9) and states with divided government (8.6 vs. 3.3), but not states controlled by the Democrats (6 vs. 5.9).
Obviously warm weather doesn't make people sick (in fact, research suggests these kind of viruses should spread slightly less in warm weather). But it might be correlated with certain types of behaviors that can lead to a higher rate of infection. No idea which though.
Interestingly, there is no link between Trump's vote share in 2016 and infections today, while having a GOP state leadership is a consistent predictor of new infections. That's another data point in favor of state policy (rather than demographics) having an impact.
For the graphically inclined, here are scatter plots that look at average summer temperatures and new Covid19 infections based on which party controls the state government (GOP control in first graph; Democratic control in the second).
The @nytimes has a list of states where new covid cases are increasing, decreasing, and staying the same.

Let's see how that breaks down by party control over state governments.

Increasing: 7 D, 16 R, 6 neither.
Decreasing: 4 D, 3 R, 4 neither.
The same: 4 D, 2 R, 4 neither.
That's to say that covid cases are increasing in 76% of the states controlled entirely by the GOP, 47% of the states controlled by the Democrats, and 43% of the states with divided government.
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