Why does the custard in my lemon/lime bars always come out looking terrible and with bad texture?

It is very difficult to tell what is wrong definitively from your pictures, but the custard appears to be curdled, or have too much air, or both.

When you make the lemon or lime curd, you don't want to mix air into it, which will cause it to puff up, then collapse.

You also don't want to overcook it, as it will curdle like scrambled eggs. You haven't mentioned whether the curd is made stove top or baked in the oven on top of the crust layer. From the look, I am guessing they are baked, and were probably over baked. Lemon curd wants to be cooked to about 170 F (77 C), which is a little lower than a standard custard, due to the high level of acidity making the proteins set at a lower temperature.

I agree that it is badly curdled. Custard per se is finicky about curdling, it has an about 10 degrees celsius "right" window - if it doesn't reach that temp, it doesn't set, if it overshoots it, it curdles. Lemon custard is much worse, because the acid curdles proteins even without high temp.

The tricks you can use: 1) warm everything really slow. It is a little-known fact that the speed with which eggs are heated determines the temperature at which they will curdle (source for this counterintuitive statement: Cookwise, a book by the food scientist Corriher).

So, leave your eggs out overnight before starting, so you will be sure that they are room temperature and not fridge temperature. Then set the oven very low. It could be that an experienced cook can get away with 350 as in the recipe, as they may be able to pull it out quickly. You will be on the secure side if you set it to maybe 230 and then bake by internal temperature, as Saj suggested.

Also, if you still experience trouble, you could increase the amount of sugar in the recipe. This will reduce the chance of curdling too.

I don’t think I've ever seen a recipe that says to whip the custard after cooking it. That might have something to do with the thickeners not binding properly, and the air bubbles falling from the weight. It also is curdled. I would try keeping the custard just on the stovetop next time. As for cooking time, bring it to a boil for about 30 seconds, but no more than a minute, and then pour onto your shortbread.

I just tried a new recipe, and it suggested straining the mixture, but my custard didn't set all the way, and I think the straining might have done that.

Anyway, test one change at a time and see what happens.