regression by grouping the dependent variable

I have a large dataset exploring the effects of the independent variables on the dependent variable using Poisson regression since the dependent variable is a count variable.

However, the range of the dependent variable is too large. Hence, I was thinking of grouping the dependent variable, like low, medium and high values, and then use the Poisson regression for each group. My question is, does this makes sense? I mean, grouping the dependent variable and then running the tests separately for each group.

One question may be what is a low, medium and high value. I noticed some clustering algorithms and I (have used) and will use one of them(k-means seems to be fine) to divide the dependent variable into three groups.

My initial analysis shows that the effects of independent variables are different for each group, but I am not sure if this is the correct way of doing the analysis.

Can you please comment?

Thank you,

EDIT: There are 454270 samples is the dataset and the range of the dependent variable is between 0 and 160 (with mean 1.293136 and std 2.681311) and 90% of the values are in the range between 0 and 3. The histogram of the variable is as follows:

A more detailed histogram of the dependent variable and two scatter plots of the IVs and the DV is as follows (there are also other independent variables but they are control variables)