I really disagree with this. I've spent time optimizing web apps (both client and server side), and I've never used bit-twiddling to do it. Your application will be very fast and have none of that.
The performance problems you run into on the server side are generally related to bad data access patterns with an external database or caching layer. They are solved by writing better queries, or organizing the data more efficiently. As an end user of those systems, you do not interact with the raw bytes yourself. Of course if you're actually writing a database engine or a cache, then you would.
On the front-end, it's a bit more complex. Often what makes apps slow is actually the way their assets are loaded. For example, using web fonts can incur a major performance hit if you're not very careful. Many shops have moved away from using web fonts for this reason. Similarly, loading javascript at the wrong time can cause things to be slow, because the browser has to wait for the JS to be loaded. Beyond that, slowness is often about inefficiently rendering data, or again loading more data than you need. To make a fast app, you mostly need to be aware of how browsers load and render JS, CSS and HTML. Bit twiddling is not really relevant.
But really what this comes down to is that no one wants to pay for performance. An app that's kind of slow still pays the bills, and engineers are not in the driver's seat to prioritize performance. Instead, the focus is on finding ways to get people to pay you more money. It's much easier to sell features than performance, so that's where the focus is.