Recurring problem with retrospective data collection study designs I'm seeing I've noticed a lot of medical research that I am involved in goes as follows:
Collect data on 300-1000 patients, including all sorts of baseline characteristics such as BMI, age, gender and then outcome related statistics, so say our outcome is "fracture after operation", we could have angle of fracture, fracture density, pain scores, mobility scores, quality of life scores etc. etc. and then finally our outcome, whether or not the patient had a fracture after the operation. Often these outcomes are binary and the goal is to see if any of the independent variables are associated with fractures.


*

*Now the problem here is we have a binary outcome variable and we
often end up with about 30-50 patients who actually had a fracture
out of 1000 patients, so the statistics are quite skewed and a lot
less powerful than if 500 of the patients had fractures.

*The 2nd problem is we have maybe 50 independent variables of diverse
types, factors, continuous, binary (am I correct to assume that in these cases p>N due to the outcome variable only encompassing, say 30 patients, even though the study size N is 1000?)

*The 3rd problem is these are often studies made with little previous
knowledge on the subject, so it's often hard to manually pick
confounders by expert opinion.
Obviously we can't run a large multiple regression with all variables as the model overfits. We can't run 50 (independent variables) multiple regression analyses controlling for say age and gender, because we quickly run into a very grim multiple comparison problem.
We can't use regularization models because we are interested in all 50 variables and whether they are associated with our outcome (none are deemed simply controls, which regularization models choose from but do not necessarily add to the model).
From a statistical viewpoint, what would be your way of handling such a study design? Currently I just run logistic regression models controlling for patient characteristics and am transparent with the fact that the p-values are unadjusted. 
I should note that these studies are not meant to invent a new method of treatment or change protocols, they are used to see what variables are of interest for future research.
 A: You are right that this is a very common scenario in medical research.

"I should note that these studies are not meant to invent a new method of treatment or change protocols, they are used to see what variables are of interest for future research."

OK, I take this to mean that you are interested in causal inference, not in prediction.
And from the comments:

"We have statisticians available to us. They suggest pooling variables with P<0.2 from a univariate regression of all variables, into a new multiple regression and report the variables under alpha in that 2nd regression model." 

This is not advisable. For one thing, mediators will be associated with the outcome, which you should not be adjusting for. You might also end up adjusting for colliders and actually invoking otherwise non-present confounding. See here for things that can go wrong when including variables that have no business being in a regression model.
I am sorry to say that there is no substitute for expert knowledge about the subject matter when it comes to causal inference. It is really as simple as that. "Expert" is  relative term. You don't have to have a PhD in the field. I thought I read in another post that you are a medical doctor nearing the end of your training. I would have thought that you would be able to come up with a plausible DAG for many scenarios. I have been involved in teaching these things to undergraduate medics for a number of years and I usually find that they are able to construct plausible DAGs quite well. It is normal for different people to come up with different DAGs because they make different abstractions and assumptions about the data. Also, when they are completely stumped they are usually able to find information online or from other resources to help and inform their DAGs.
