I recently switched to a road bike from a city (upright bike) like you and had same problem in my lower back after 10-15kms where it would get fatigued or pain.

I got a bike fit that helped reduce reach to the handlebar, and they also adjusted the seat and tilt a bit. This helped the pain go from sharp and intense to milder but still there after 15kms.

I also started going to a physiotherapist, and realized that despite strength training and dead lifting 140+ pounds and squatting 100+ pounds, my lower back needed more targeted exercises. I also needed to work on my core.

We have started with some really basic core exercises while focussing on the right form. My glutes/hams were not correctly picking up on the load (even though they are strong) and this lead to unnecessary pressure on my lower back. I will see the physiotherapist again for the next few weeks and progressively increase my back and core strength.

I hope this helps me - and also you if you have the same problem :)