Cycling involves two types of muscles: slow twitch and fast twitch (
https://en.wikipedia.org/wiki/Myocyte#Fiber_types).
Fast twitch muscles develop quickly from exercise and are what you use for quick bursts of speed and what cyclists call "power".
Slow twitch muscles require a lot of blood supply and growing new blood vessels takes time. You need to be patient with climbing as you are literally growing new blood vessels to supply your fatigue-resistant muscle fibers.
I would give it a year of steady improvement myself. You will get there.
Or you could get lower gears and do like me: realize that your purpose in life is to give everyone else someone to pass.