How do I turn off read receipts on my iPhone so people can't see I've read their texts?

You've got two levels. For everyone: Settings > Messages > toggle off Send Read Receipts. Now nobody sees your 'Read' on iMessages. For just one person: open the chat with them, tap their name or photo at the top, and toggle Send Read Receipts off there, which overrides the global setting for that one thread. Note this only affects iMessage (blue bubbles). Green-bubble SMS never had read receipts, so there's nothing to turn off there.

The per-person toggle is the underrated one. Leave read receipts ON for your partner so they know you saw them, and OFF for everyone else. Tap the person's name at the top of the chat to find it.

Just remember the global setting is mutual, turn yours off and you also stop seeing when others read yours. The per-conversation toggle avoids that.