How would profit from labor exploitation be possible in a relatively competitive market? Marxists often state that capitalists exploit workers by paying them less than than their labor is worth. The extraction of this surplus value results in profit.
I'm wondering why this would be possible assuming most markets are fairly competitive.
It seems like if labor is really producing so much value that profit is possible, other companies would come along and snap up those workers with higher wages and eventually profits would minimize.
But this doesn't seem to happen. Most industries are quite profitable.
And the general trend of wealth accumulating in the upper class suggests that a balance has not been reached.
I have seen it suggested that we shouldn't need minimum wage because it forces companies to pay people more than their labor is worth, but if that were true, it seems like profits should already be small.
I understand that the situation differs considerably between industries, but the overall trend seems to be that most industries are able to be fairly profitable.
Would it be fair to say that while wage labor isn't inherently exploitative, most industries have some sort of market failure (externalities that result in only a few large companies, etc...) so in practice most wage labor IS exploitative (i.e. workers are paid less than the value of their labor)?
In competitive industry it is not possible as wages will be paid their marginal product of labor, and 'accounting' profit will arise either from marginal product of capital or entrepreneurial labor.
In economics, it is important to distinguish between economic and accounting profit, the economic profit in perfectly competitive market is zero, but you cannot observe that just by looking at the accounting profit firms report as that does not take into account various costs such as opportunity costs (generally accounting serves practical purposes not to scientifically describe true underlaying economic reality - for example in many countries sole traders/self employed people will be by accounting systems allowed to write of let's say $30\%$ of revenue as a cost - this may make sense from accounting perspective but economically it is completely absurd to state that all self-employed people have such costs).
In addition, just because there are imperfections on goods market (allowing firm to earn economic profit) that does not mean there will be imperfections on labor market (meaning employees will still get their marginal product).
Lastly, even when labor market is imperfect, and these imperfections result in labor getting less then its marginal product (there could also be imperfections that would result in more) there is no exploitation going on in the Marxian sense of exploitation.
The Marx's theory of labor exploitation can be summed as (see Stanford Encyclopedia of Philosophy):

To understand Marx’s charge of exploitation, it is first necessary to understand Marx’s analysis of market prices, which he largely inherited from earlier classical economists such as Adam Smith and David Ricardo. Under capitalism, Marx argued, workers’ labor power is treated as a commodity. And because Marx subscribed to a labor theory of value, this means that just like any other commodity such as butter or corn, the price (or wage) of labor power is determined by its cost of production—specifically, by the quantity of socially necessary labor required to produce it. The cost of producing labor power is the value or labor-cost required for the conservation and reproduction of a worker’s labor power. In other words, Marx thought that workers under capitalism will therefore be paid just enough to cover the bare necessities of living. ...


But while labor power is just like any other commodity in terms of how its price is determined, it is unique in one very importance respect. Labor, and labor alone, according to Marx, has the capacity to produce value beyond that which is necessary for its own reproduction. In other words, the value that goes into the commodities that sustain a worker for a twelve-hour work day is less than the value of the commodities that worker can produce during those twelve hours. ...


According to Marx, then, it is as though the worker’s day is split into two parts. During the first part, the laborer works for himself, producing commodities the value of which is equal to the value of the wages he receives. During the second part, the laborer works for the capitalist, producing surplus value for the capitalist for which he receives no equivalent wages. ...


Capitalist exploitation thus consists in the forced appropriation by capitalists of the surplus value produced by workers.

As you can plainly see above the idea of exploitation rests on the idea of labor theory of value (LTV). Labor theory of value is no longer accepted theory since it leads to empirical inconsistencies and it is generally considered incorrect (the current accepted theory of value is subjective theory of value). Under subjective theory of value anything can create value, including capital and indeed as we are seeing more and more automation that eliminates the need for labor this is becoming more painfully clear than ever.
In addition, the Marx's analysis implies that industries which are more labor intensive should have higher rates of profit (since according to Marx profit is earned through exploitation) and this empirically does not hold (Böhm-Bawerk 1898).
Next, the whole theory on which Marx build contained analytical mistakes that were not resolved by Marx (see Grant &amp; Brue The History of Economic Thought 7th ed).
Lastly, the Marx's theory of exploitation also relied on an argument that workers are forced to work for capitalist as they don't have access to capital. That could seem reasonable in context of early 19th century England, when capital was very expensive and incomes low so for most part only richer people could afford to invest in it, and thus they were 'forced' by circumstance to work for capitalist.  However, this is not applicable when workers have access to out option (welfare benefit) as they often do in modern countries. In addition,  with modern financial systems the access to capital if you have good idea is arguably better than in 19th century where capital markets were still developing.
Hence as pointed by KennyLJ in the comments the Marx theory of exploitation is discredited in the profession. There were some scholars who tried to 'resurrect' the Marx idea of exploitation by dropping the labor theory of value and arguing only some of the 'worker surplus' is being exploited, and also trying to argue it applies to situations where workers are not forced to work  (see Kymlicka 2002, Contemporary Political Philosophy: An Introduction), these ideas are not generally accepted in economics and rather live on in some corners of philosophy/sociology.