For these sentences:
- The road goes around by the lake.
- The road goes around the lake.
- The road goes by the lake.
How are they different?
1) The road goes around by the lake.
2) The road goes around the lake.
3) The road goes by the lake.
All three versions are fine, but they each can sometimes have different connotations. And in many different contexts, different speakers might prefer one version over the other.
Let's first look at some possible differences between versions #2 and #3. Here are some edge cases where, normally, only one of the two versions would be acceptable.
Edge case for:
Suppose a summer camp has a lake and there is a road that circles that lake, and that is all that road does. For that kind of road, a speaker would say: the road goes around the lake. A person would not normally say that the road goes by the lake. (Though, it is true that at any one spot in the road, one could logically think, or maybe even say, that the road goes by the lake.)
Edge case for:
Suppose you are driving on a highway, and the highway basically goes in a straight line. On the map, it looks like there is a lake a short distance away, but the highway doesn't bend toward it or curve around it at all. And so, a person would say that the road goes by the lake. Since the road never bends toward or curves around any part of the lake, a person normally would not say that the road goes around the lake. (Note: In general, version #3 could have a meaning similar to "The road goes past the lake.")
Okay, those were two edge cases where, normally, only one of the two versions would be acceptable.
But when the road does not fully circle around a lake, such as when a road curves around only part of a lake even though the road might be some distance from the lake, or when the road bends toward a lake (on a map), then many speakers might choose one version or the other, or both. That is, the road goes around or goes by the lake, as the road goes from its source to its destination.
Many speakers might also choose to say the remaining third option:
The speaker might do this in order to indicate that the road curves around part of the lake, and that the road is relatively close to the lake when it does that. But as to how much the road must curve or how close it must be to the lake, that is a subjective matter and depends on the speaker. (There's also another possible context when this version could be used: when the road goes past a lake when it goes around something else that is understood. That is, the road goes around something via the lake.)
Version #1 usually would not be used for either of the two edge cases that were discussed at the top of this post. (Though, a speaker might use it to indicate that the road circles very closely around a lake: that is, the road goes around by the lake.)
So, hopefully the gist of this post is somewhat clear enough. That is, there's a lot of overlap between the three versions, and other than the edge cases, the speaker decides which version is preferable in a given context.
(Note: AmE speaker wrote this post)
2) is OK. It is as written.
1) might not be technically wrong, but around the lake would imply it is by the lake, so it is redundant at best.
It would be more natural with some additional details, for example:
The road goes around the park by the lake.