This is my workaround on extending multiple classes. It allows for some pretty sweet type-safety. I have yet to find any major downsides to this approach, works just as I would want multiple inheritance to do.
First declare interfaces that you want to implement on your target class:
interface IBar {
  doBarThings(): void;
}
interface IBazz {
  doBazzThings(): void;
}
class Foo implements IBar, IBazz {}
Now we have to add the implementation to the Foo class. We can use class mixins that also implements these interfaces:
class Base {}
type Constructor<I = Base> = new (...args: any[]) => I;
function Bar<T extends Constructor>(constructor: T = Base as any) {
  return class extends constructor implements IBar {
    public doBarThings() {
      console.log("Do bar!");
    }
  };
}
function Bazz<T extends Constructor>(constructor: T = Base as any) {
  return class extends constructor implements IBazz {
    public doBazzThings() {
      console.log("Do bazz!");
    }
  };
}
Extend the Foo class with the class mixins:
class Foo extends Bar(Bazz()) implements IBar, IBazz {
  public doBarThings() {
    super.doBarThings();
    console.log("Override mixin");
  }
}
const foo = new Foo();
foo.doBazzThings(); // Do bazz!
foo.doBarThings(); // Do bar! // Override mixin