Bugs: How to reduce the number

Recently the team I'm part of finished a major feature that had so many bugs it was painful. We don't have a QA team. Looking at the bugs produced by me, they were either lack of testing or missed requirements.

There's so much pressure to get stuff done quickly. I'll often get conflicting demands, like: Being told to put a task aside and then weeks later the same manager will demand it be pushed out asap, or being told to take more time to review others' PRs, yet being told I need to get mine reviewed quicker, or being told to test in our test environments… that we share with our clients.

I don't really know what to think, I'm starting to feel like a crappy dev, but also feeling frustrated by the demand to produce perfect code quickly, whilst testing everything else and reviewing everyone's PRs, coaxing reviews on mine, spontaneously refactoring, tracking xyz, blah gah etc.

How do you guys manage it? What do I need to do to get better?