WASHINGTON—Mutual funds will soon face new regulations to reduce the risks of an investor exodus during a panic, though they have managed to win some concessions that will make the rules more flexible.

The Securities and Exchange Commission is set to release the new rules Thursday, a key part of its plan to modernize supervision of the $15.7 trillion sector. The once staid business of pooling stocks and bonds has become more complex and risky, as funds adopt strategies that involve harder to sell securities, derivatives and...