I decided to investigate your claim because it smelled fishy - and yeah, you're wrong.
Foreign born women do have slightly higher birth rates than native-born women, but it's nowhere close to 4-5 kids: between 2000 and 2010, native-born women had, on average, around 1.7 children each, compared to foreign-born women having on average, around 2.1 children each. However, because foreign-born women make up comparatively few of the Swedish population, this effect only increases the average birth rate by around 0.1 child:

Furthermore, to hammer my own point home, income quartile has a much larger effect on birth rates, with the difference between income quartile 1 (0.9 births avg) and income quartile 4 (2.4 births average) having a much stronger effect:

So, yeah, it's not video games, and it's not really a migration issue. As I said in my original reply, it's a lot more to do with financial security and stability.