TL;DR: I have no full solution, even now 8 years after question was asked. But you can find a browser plug-in that does this for you.

Trafitional web browsers for desktop just do what you requested: when changing the zoom level, they rewrap (re-flow) displayed text to be readable without scrolling horizontally, which is the most accessible way you could consume text.

However, with some developments this native feature got more and more forgotten:

• Styled web pages (and mail) with fixed widths: many web designers started to give web pages a fixed width, irrespective of browser width.
• Later, with responsive web design, it got way better on many places: desktop browsers started wrapping again and mobile browsers picked a reasonable initial font size, while still not doing the re-wrap.
• Mobile web browsers and the idea of viewports which focuses on your mobile phone acting as a simple window scrolling through a webpage. Two-finger zooming only resizes that window and does not effect the page's dimensions. The idea might have been that re-flowing was expensive in CPU power, or also jumping HTML content when text would be re-flowed.

My opinion is that it's a shame there is still no (satisfying) answer to this (that I know of), because I think that zooming and constant horizontal scrolling is a severe accessibility issue of many webpages and e-mails when consumed with limited sight on mobile devices.

Some web designers respond to this lack of accessibility by placing buttons to render their web page in several font sizes (as also explained in Bob's comment), but that option is often not found.

However, the good news is that for some mobile web browser there are browser plugins that do what you need. Not being in browser core, however, makes them less stable.

One example is the Firefox / Fennec plugin: Text reflow (on mobile) it works well for me on most web pages that are not too strictly styled.

You might need a different plugin for your web browser of choice (or any other app displaying HTML content, like e-mail app)