How can I copy the content of a branch to a new local branch?

See second part (since Git 2.23, Q3 2019): git switch -c newBranch oldBranch

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With Git 2.15 (Q4 2017), "git branch" learned "-c/-C" to create a new branch by copying an existing one.

See commit c8b2cec (18 Jun 2017) by Ævar Arnfjörð Bjarmason (avar).
See commit 52d59cc, commit 5463caa (18 Jun 2017) by Sahil Dua (sahildua2305).
^{(Merged by Junio C Hamano -- gitster -- in commit 3b48045, 03 Oct 2017)}

branch: add a --copy (-c) option to go with --move (-m)

Add the ability to --copy a branch and its reflog and configuration, this uses the same underlying machinery as the --move (-m) option except the reflog and configuration is copied instead of being moved.

This is useful for e.g. copying a topic branch to a new version, e.g. work to work-2 after submitting the work topic to the list, while preserving all the tracking info and other configuration that goes with the branch, and unlike --move keeping the other already-submitted branch around for reference.

Note: when copying a branch, you remain on your current branch.
As Junio C Hamano explains, the initial implementation of this new feature was modifying HEAD, which was not good:

When creating a new branch B by copying the branch A that happens to be the current branch, it also updates HEAD to point at the new branch.
It probably was made this way because "git branch -c A B" piggybacked its implementation on "git branch -m A B",

This does not match the usual expectation.
If I were sitting on a blue chair, and somebody comes and repaints it to red, I would accept ending up sitting on a chair that is now red (I am also OK to stand, instead, as there no longer is my favourite blue chair).

But if somebody creates a new red chair, modelling it after the blue chair I am sitting on, I do not expect to be booted off of the blue chair and ending up on sitting on the new red one.

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Second part: with git 2.23 (Q3 2019), no need to use git branch or the old confusing git checkout: you have git switch.

git switch -c newBranch oldBranch

git checkout old_branch
git branch new_branch

This will give you a new branch "new_branch" with the same state as "old_branch".

This command can be combined to the following:

git checkout -b new_branch old_branch