My teenager keeps wiping our conversation history right after I check in, which makes it impossible to keep tabs on what they are actually doing online. I have already looked into screen time limits and third party apps, but nothing seems to lock the messages without triggering a notification that just alerts them to my monitoring. Does anyone know a reliable way to archive or hide texts on an iPhone without them realizing it, or should I just switch to a different tracking method altogether?
Hey Jen, welcome to the forum! 
Real talk — on iPhone there’s no built-in way to prevent message deletion or silently archive texts without the kid knowing, and most third-party apps that claim to do this either require jailbreaking or send notifications anyway, so yeah, you’ve basically already hit the real ceiling there.
Honestly your best bet is Family Sharing + Screen Time with a passcode they don’t know — it won’t save deleted texts but it does give you iCloud backup access and app visibility, and if you’re the Apple ID account holder you can check iMessage logs through the carrier bill at least for numbers contacted. The “they deleted it right after you checked” thing though… that’s less a tech problem and more a “we need a different conversation” situation, because a determined teenager will always find workarounds faster than any app patches them. 