No, a 24mm lens on a 24x36mm sensor will draw the same angle of view as a 16mm lens on a 16x24mm sensor. Yes, this is a very wide lens, and it is hard to control the funkiness that results from using such a wide angle of view. I remember reading about a classic street photographer (Winogrand?) who wanted the widest possible lens, but settled on a 28mm not because it was the widest but because it was as wide as he could get with a reliably appealing drawing for his purposes. So, don't assume that just because some photogs prefer 24mm means you should, too; 28mm and 35mm are also quite useful for environmental portraiture (as can be 18mm and 21mm).
Without seeing what you're talking about, it is hard to tell you what you're seeing. But, you might be talking about perspective distortion or areal distortion.
Test: take an image that shows strong distortion of the kind you're seeing. Edit it in a program that allows you to correct for distortion. Now, add barrel distortion (or tell the program that you want to correct for pincushion even when lines at the edges are already straight). This will reduce areal distortion. If this solves your issue, then you know what it is.
Theory overview--
I think there are three kinds of distortion:
1 - Perspective
2 - Linear
3 - Areal
Perspective distortion is the result of the distance between objects and the camera. Lenses do not affect perspective distortion beyond making certain things possible or in how a particular lens facilitates the use of particular types of shots.
Linear distortion is usually what people mean when they're thinking of lens-related distortion. Take a picture with a flat subject along one edge of the frame; if straight lines are bent, you are seeing linear distortion.
Areal distortion is usually what people are referring to when they say wide lenses distort people's heads. Of course, it distorts all subjects, but we're just more attuned to the way a human head naturally looks so we're more sensitive to distortion of it. Take a picture of a regular subject (say, a round sign) in the corner of the frame; if it appears like an egg shape rather than a circle, you're seeing areal distortion.
Linear distortion and areal distortion are directly and inversely correlated. Eliminating linear distortion means accepting areal distortion and vice versa.
Thought experiment: consider a brick wall of infinite length and width. Each brick is the same size. But, if you look to one side, the bricks further away appear to be smaller (due to perspective distortion). Now, try drawing an image of a brick wall where bricks toward the edges of the frame are smaller than those in the center. The accurate result is barrel distortion. The wider the angle of view, the stronger the naturally occurring linear distortion. In order to eliminate that distortion, the bricks at the edges of the frame must be drawn larger than they actually appear. This distorts the appearance of areas, creating areal distortion.
In my experience, wide-angle lenses that are perfectly corrected for linear distortion can be rather ugly in some scenes due to areal distortion. Adding a hint of barrel to the image makes it look better. Older, classic lenses do this intentionally: it isn't that optical design has only recently allowed for perfect rectilinear designs but that, before the rise of internet reviews and "oh, curved line, must be bad lens" comments, lens designs balanced linear and areal distortion. (This might be important if you're looking at photos taken with 24mm lenses from the 1980s or earlier, as they weren't as strictly barrel-free as today's designs.) Fortunately, some software-correct lenses offer the best of both worlds: the native image has quite a bit of distortion, an "ideal" correction for straight lines is very easy to achieve, and the photographer is able to decide where in the middle each image should fall.
Hope this helps,
Jon