Lately the first thing that "gives out" - gets tired with a burning feeling on a long ride, are the sides of the hips.
I'm thinking upper part of gluteus major, TFL or whatever else may originate from around the iliac crest.
The same area is also what is usually the most sore after a longer run.

Now that I have become a convert to gym training:
What exercises?
I am already being a good girl with leg raises in the half side plank position and squats/split squats, as directed by my PT to protect the ITB.