Can I watch movies or web series that contain some inappropriate content but aren't fully haram?

This is a genuinely divided area rather than a single rule. A stricter scholarly view holds that a film or show containing clearly haram elements — nudity/immodesty, music (per scholars who view music as impermissible, see Q2), glorification of sin, or content mocking religion — should be avoided in full, even if surrounded by otherwise acceptable content, on the principle that one shouldn't expose oneself to haram content as entertainment, and that "mostly fine" doesn't offset genuinely objectionable scenes. Some scholars extend this to avoiding an entire show or franchise once it's known to regularly include such content.

A more contextual view, held by other contemporary scholars, distinguishes between content that is incidentally present (a brief scene, background music, mild content that isn't the show's focus) and content that is the show's actual purpose or dominant theme (explicit romance, glorified violence/crime, anti-religious messaging) — under this view, occasional or skippable inappropriate content in an otherwise beneficial or neutral show is a lesser concern than actively seeking out content built around haram themes, though caution, minimizing exposure, and being honest with oneself about the effect on one's heart and habits are still strongly recommended either way. Given the range of views, this is an area where personal discipline and knowing your own weaknesses matters as much as the abstract ruling — several scholars explicitly frame it as a question each person needs to answer honestly for themselves rather than a fixed universal line.

Source: SeekersGuidance, "Is It Unlawful to Watch Movies or Read Novels?"; Islamweb, "Watching Movies That Include Prohibitions if Convey a Good Message".