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A steamboat captain trying to save his sidewheeler a few miles made a little improvement to the Mississippi River 156 years ago. Over the decades, it has cost United States taxpayers nearly $500 million for repairs.

But now the Army Corps of Engineers says its latest effort to fix things, a $220 million structure, takes the weakness out of the river curve once called Turnbull's Bend.

In the heyday of steamboats, Turnbull's Bend was a long, lazy meander about 220 river miles upstream from New Orleans. It took a boat hours to travel its 20 miles, and then it was only a mile or so farther along the way.

In 1831, Capt. Henry Miller Shreve, who founded Shreveport, La., brought in 159 men to dig a ditch across the neck of land almost encircled by the curves. At the next high water, the Mississippi roared through.

Shreve's ditch rearranged the natural order. The Red River, which had flowed into the Mississippi, began emptying into the smaller Atchafalaya. The Atchafalaya, which flowed out of the Mississippi at Turnbull's Bend, was now connected at the abandoned loop that came to be called Old River.

Force-fed by the Red and the Mississippi, the Atchafalaya wore deeper, wider. And then the Mississippi took over the Atchafalaya's 142-mile bed.

Turnbull's Bend is the point of maximum stress along the river's entire 2,340 miles, engineers say. It is where the Mississippi keeps trying to chew through. If that happens, a few small towns could be inundated and much of the river stretching from here past New Orleans would become a salty estuary of the Gulf of Mexico. The mouth of the Mississippi would move west 120 miles to Atchafalaya Bay.

The new project, scheduled for dedication today, was built to ease stress on an Old River control structure built in 1959 but partially undermined by floods 13 years later. Both structures are, in effect, dams between the Mississippi and the mouth of Old River. They control how much of the Mississippi flows into the Atchafalaya.

Whether the Corps of Engineers can keep a tight grip is a matter of dispute.

Ralphael G. Kazmann, a retired engineering professor, says that, in the long run, Old River cannot hold.

To Leroy Dugas, the project engineer, the key factor is money. ''If Congress gives us the funds to do the work, we can hold,'' he said.

But David B. Johnson, an economics professor at Louisiana State University, thinks the decisive factor will be whether holding the river is worth the ever-increasing expense.

In 1980, Mr. Johnson and Mr. Kazmann published results of a two-year study concluding that the Mississippi will win. They did not predict when; they just said it might be when the river, swollen by rain and melting snow from 31 states and two Canadian provinces, is running fast and mean.

The Kazmann-Johnson study estimated 140,000 people would have to flee the Atchafalaya Basin if the Mississippi broke through.

One voice that has not been raised in alarm is that of Oliver Houck, a Tulane University professor of law specializing in environmental matters. Threatened towns could be protected, Mr. Houck said, and the ports of New Orleans and Baton Rouge might be better off if the Mississippi came through. If the Mississippi reached the sea somewhere else, the rich silt would create a new delta, he said.

Of course, no one knows what Shreve would say about all this.