Many have already made excellent suggestions regarding transforming the variables and using robust regression methods. But, when looking at the scatter plot, I observe two separate data sets. One set has a very strong linear relationship where the correlation is a lot higher than the overall 0.6. And, visually it looks like Y = 0.13X. So, when X = 15,000 Y is around 2,000 or so. Thus, a regression line with a similar slope would fit the vast majority of the data points really well. Then, you have a second data set of 300 datapoints that are wild outliers that are random.
I would focus on those 300 outliers. Can you explain them? Are there reasons why they are so far off the regression line? Are those datapoints a fractional % of your whole data set? Are they material events you need to keep for your study? Or can you afford to take them out? If you can take them out, you may have a pretty strong regression with a high R Square. You just would have to accept that in a few percentage of the time things go wild and your regression model will be off. But, that's the truth of any model you built.
If you have to keep those 300 outliers in your overall data set, they will materially affect your regression. And, you will end up with a regression model that does not fit well the majority of your data point. And, it won't fit the outliers either because they are random and won't fit any regression line.