Because the revenue they generate is massive. It's impossible to lose money owning a team in one of the three major sports. NBA clubs have notoriously lost money for years, to the point of two work stoppages in the past 15 years. You're leaving out the NHL despite the fact they are often touted as one of the four major sports and is the biggest league worldwide for its sport - they've lost massive amounts of money. MLB clubs lost money in decades past but appear to be much healthier today. NFL clubs were losing money until they forced a CBA that imposed a cap in 1992 (effective 1994). Smart management should avoid losing money with the massive revenues US sports clubs generate but not all clubs are run fiscally smart. The Dodgers and Angels are spending lavish amounts and have nothing to show for it, possibly operating in the red as we type. They can do such with their local TV contracts and the big market advantange, much like Barca and Real in Spain. However, unlike the Spanish clubs these MLB clubs will fork out a large luxury tax and do not have the global fanbase to generate millions in additional revenues.
NFL players get graphically maimed every week for our entertainment and many of them go bankrupt. Teams refuse to renegotiate contracts of underpaid players yet callously cut overpaid ones. For many players yes, however, star position players do get paid very well and many have their contracts restructured for various reasons. QBs always get their money, so do top tier players at other positions. No one has ever put a gun to the head of a professional athlete so he assumes the risks of his sport. Athletes also require lesser years of service to qualify for pension and other benefits. Not a single athlete in the four major US sports earn less than 300k in a given year (for a full season; rightfully structured for actual days/weeks active if signed during the season).
Real Madrid are paying $100 million for Bale plus whatever his wages are. It's not like they've decided that the money invested is too high for a player of his caliber, they just allocate a vast majority to the selling club. If there was no selling club, the economics don't change. If they've valued the acquisition and signing of Bale for five years at $125 million then in a situation with no selling team, more of that would be given to the player. US sports do not pay transfer fees so it's impossible to compare to other continental sports leagues. The entire business of negotations would change if transfer fees were eliminated (ex: MLB had buy/sell clauses many decades ago until trades became the norm). Clubs may amortize costs of transfers plus salary/wages and bonuses simply for budgetary reasons. Transfer fees also do not relate to actual market value - as we've seen with the sugar daddy clubs and with the big revenue clubs that can overpay to acquire a player. Institute a salary cap and/or transfer cap and we would see transfer fees drop collectively, IMO. In no sense of reality is Bale worth €100m transfer fee. To Real they can simply afford a massive fee where as 99.9% of the other clubs cannot imagine spending that on one player (I doubt even City or PSG would, except for TBPITW which Bale is nowhere remotely near the level of). I can't begin to imagine how the sports in the US would operate if they used a transfer fee system.
Did you see the TV deal the Dodgers got? They bought the team for $2 billion (which many thought was an overpay) and then turned it into $25 billion of TV rights before selling a single ticket, jersey or hot dog. The players generate gobs and gobs of money for the owners, it's only fair that they be compensated for that. True, and we all know the Dodgers are not worth $2.15b paid and are nowhere near the global entity like the Cowboys and Yankees (leaving out the obvious European football clubs). The deal included 250 acres of land and the stadium. 250 acres of land in Southern Cal is priceless. The biggest reason there was a massive bid (next highest bid was $1.5b) is the lure of a regional sports network which is potentially worth billions. If this happens, the SoCal sports fan might be shelling out monthly fees to see his/her fav sports clubs on a regional network. This is why Mark Cuban was looking to buy the Rangers - to form a regional network with two sports clubs.
Players do get royalties from the league and product manufacturers for merchandising rights. Some players have separate endorsement deals. The sky-high salaries have resulted in the rising costs to attend games including ticket costs, concessions, merchandising, even ridiculous parking costs, though I will concede that it's quite likely owners claim the salaries as the biggest factor but I'll say there is greed involved as well. Us fans are the fools for paying such amounts. We're addicts.
Every team in the league was lining up to give Lebron $20 million, you don't think the Nets, Lakers and a couple of others wouldn't roll out the red carpet with $75 million offers if there were no salary cap? Anything is possible but I highly doubt a club is going to pay a single player $75m per year. Perhaps in a few more decades if revenues and salaries keep rising. The Nets are certainly the exception just like City or PSG.
As for Messi, the impact of a basketball player is higher than the impact of a footballer, just because of the structure of the games (roster sizes, players active, etc). But if Messi was a free agent right now don't you think he could get $50 million per year? I bet he could blow past it.