Carrier addresses Josephus, both in the On the Historicity of Jesus [0], and in his blog posts [1]. His points, as I understand them, are that: 1) The first Josephus passage — the so called Testimonium Flavianum — is an insertion by Christian scribes. He gives stylistic arguments for why this is the case: both by how dissimilar this passage is from Josephus's style, and by how similar it is to the Gospel of Luke; and 2) In the second passage, which mentions James, "the brother of Jesus, who was called Christ", the reference to Jesus had originally been a marginal note by a Christian scribe that subsequently got interpolated into the main text.
This idea of an interpolated marginal note is Carrier's own contribution, first published in the peer-reviewed Journal of Early Christian Studies in 2012. While elegant, is not the only explanation. In his blog [1], Carrier references several recent publications (peer-reviewed, as he likes to emphasize), including one published in Journal of Early Christian History in 2017 by a South African historian Nicholas Peter Legh Allen, who argues that the James passage is a forgery, of which his prime suspect is Origen [2]. So, you can see that Carrier isn't the only one among modern academic historians of antiquity who casts doubt on Josephus passages.
---
[0] Josephus and the Testimonia Flaviana in On the Historicity of Jesus, 2014, p. 332-342
[1] https://www.richardcarrier.info/archives/12071
[2] Abstract at https://www.tandfonline.com/doi/epdf/10.1080/2222582X.2017.1.... Full text can be found in: Christian Forgery in Jewish Antiquities: Josephus Interrupted, 2020, of which a pirated pdf exists online