Birches aren't white throughout. You got the darkening of the trunk towards the bottom, but you missed that the thin parts of the branches are a reddish black.
Look how little white there is
in this photo.
Birches also usually taper much more slowly. A birch with a trunk that thick at the bottom would be much taller, and a birch of this height would have a thinner trunk. Trees with thicker trunks also usually have a more substantial dark area at the bottom.
Birches are also structured a little differently, and structure contributes a lot to whether a tree looks "right" in a drawing. Their branches usually grow more upwards (higher then 45 degrees, often higher than 60, but it depends on the environment, e.g. how many trees are around it) and then bow away from the tree because of gravity, and sub-branches are usually also not angled that far away from the main branch.
The branches are usually not so straight, either. They "weave" a lot, as do the trunks. In birches, every substantial branch causes the trunk/source-branch to grow away from it a little, so overall the branch remains "balanced":
(This pattern is common in many trees, but it stands out in birches because birches are particularly prone to splitting into two almost evenly-sized branches, instead of only having smaller branches. This happens with trunks too, there are many twinned+ birches out there.)
Your birch also has a very gradual reduction in branch size. The thinnest branches are all attached to slightly thicker branches. Real birches have many thin branches growing directly out of thicker ones, including out of the trunk.
With the texture of the bark, I think you've done pretty well! I'd put more and larger dark spots towards the bottom, but I think the main thing holding it back is that the branches are white instead of dark. Once you fix that, I think your existing texture will look pretty good! If you want to push it further, you could add more broken-up small spots, and ones that don't go full-dark.
