https://www.runnersworld.com/beginner/a20811257/proper-runni...
This is exactly how I got a bad case of shin splints. I don’t mess around with upping mileage this quickly anymore.
Too much running before the body gets strong enough to handle it, along with hard surfaces are more likely the culprit behind shin splints than poor choice of shoe.
I’ve been doing 1 miles after my strength workouts ~3x a week, and it hasn’t gotten much better. Been doing it for about 2 months.
I recommend swapping the running and lifting for a couple of months until you get that working better. Once you are running 3-5 miles then you'll give yourself enough time to switch to burning fat as fuel and you won't have so much trouble running after lifting (the first mile is always hard, though).
Or at least that was my experience. Haven't done it in more than 20 years... (stopped lifting -- kept running ;-) ).
Backstory. I used to not like running at all. Every time I ran, my mind would count down either time left or distance left. It always felt easy in the beginning but very soon would be unattainable. But now I can run decent distances without getting mentally or physically exhausted.
So here are the lessons:
1. The first 1 mile is the hardest.
2. Second mile is hard but not nearly as hard as the first mile.
3. For the first few weeks do not run more than 1- 3 miles (based on your history) even if you can. There is no need to. The habit is more important than what you achieve on any given day.
4. Run slowly, lean forward and really reduce the impact on your knees. I started running one mile in 12 - 14 minutes, that is really slow.
5. Breathing. If you run slowly and let your body do it's thing, then your breathing will become rhythmic and in-sync with how much energy your body needs.
6. After a few weeks magic happens. You can run 8 miles in about one hour, not get bored, not think about it. You mind zones out and focuses on something productive.
I learnt more things, but wanted to share this.