As far as i know logistic regression can work on non linear data , but for that we have to perform some featuring engineering like using some mathematical function to transform those features & after that we can use logistic regression .
Why logistic regression can't work on non linear data?
Yes sure, consider a two class dataset which can be separated by a parabola. If you apply Logistic Regression directly than it will be unable to find that hyperplane, but now afterwards you do feature engineering and added a new column with values x*x, now you logistic regression will be able to find the hyperplane.
But the major issue is that, you are able to do so becuase you are able to visualize this problem, what if number of features increases and hyperplane becomes more complex. So that’s the reason we depend on deep learning part.
Hope this cleared your doubt 