THE NARROW, winding road up famed Mont St. Michel to the abbey "looked like Main Street in Hyannisport on the Fourth of July," says one Washington traveler just back from two weeks in France. "There were so many Americans crammed into that tiny street."

Americans who know a bargain when they see it have been flocking to Europe in record numbers this summer, and, whether they like it or not, they are running into each other all over the continent. Busloads of Americans--and a lot of Germans, too--can be spotted in all the standard tourist haunts. In the Loire Valley, says the returned traveler, "they were herding them in and out of the chateaus like cattle."

If that's the grin-and-bear-it pain of this summer's travel, the great pleasure is in seeing the sights at a substantially better price. Not in a long time has the vacationing American had it so good in Europe.

Great Britain, the most popular destination, expects to attract 2 million Americans this year, up by about 300,000 from last year. "That 2 million figure has been a real psychological mark," says Janet Piorko of the British Tourist Authority in New York City. "We've approached it but never reached it." Her office received 239,673 travel inquiries in 1982; by May of this year, the figure was 366,184 and growing. "It's been hectic."

The European Travel Commission, which represents 23 West European countries, originally estimated a 5 percent to 6 percent increase in American travelers, but now thinks that is a conservative figure. West German and French tourist officials put the number closer to 10 percent in their countries. In 1982, an estimated 4.1 million Americans visited Western Europe.

A major reason for the surge in transatlantic travel has been the strong performance of the dollar against the European currencies, especially the French franc, British pound, Italian lira and West German mark. This summer a dollar buys as much as 30 percent more than it did last year.

One morning earlier this month a Washington shopper stopped at a jewelry store on Paris' fashionable Rue du Faubourg St. Honore to look at a ring for her mother. The price, quoted in francs, came to about $269. When she returned that afternoon to make the purchase, the dollar had improved so much on the exchange rate the new price was only $218.

Tales like this from returning travelers have whetted the appetite of bargain hunters, but a number of other factors help account for the American wanderlust: Cheaper airfares this summer (on both scheduled and charter flights), down $100 or more from last year on a transatlantic roundtrip ticket; the improving U.S. economy, which has also boosted travel within the United States; and what European travel officials see as the "pent-up demand" to visit the Old World by Americans who postponed trips when costs skyrocketed in the late '70s.

"People always want to go to Europe," says Nancy Friedman of the European Travel Commission. "The prices were so steep in the '70s, but now they can afford to go."

One good indicator of U.S. travel abroad is the number of passport applications. They are running well ahead of last year, itself a record year in which crowds all but swamped local passport offices. Better prepared this summer, the State Department reports no major problems with long lines except in the Manhattan office, where after the Fourth of July weekend the wait was more than four hours.

At the end of July (10 months into the current fiscal year), the Passport Office had received 3,621,000 applications so far this year; in the full 12 months of the previous fiscal year, the number was 3,600,000.

At least some of the travel abroad appears to be spontaneous, an indication that news accounts of the soaring dollar is spurring Americans to make last-minute decisions while the going is so good. "I would say quite a bit is being done on short notice," says Barbara Howell of The Washington Travel Agency. "Usually people plan a little further ahead." Her European-bound business "is just booming."

The people who fly the planes and rent the hotel rooms obviously are delighted.

On its daily Washington-to-London 747, British Airways reports sales from April to July ran 24 percent ahead of the same period last year. Among the airline's departure cities, "Washington is doing phenomenally well," says spokesman John Lampl. One week in mid-August, the load factor was "93 percent, which is very high." The airline's other East Coast gateways also show large increases this year.

Trans World Airlines customarily establishes its summer schedule in April and sticks to it. "But this year," says spokesman Dave Venz, "we quickly saw we had to modify it." The airline, which has a daily New York-to-Madrid flight, added a second giant 747 on the route for five days of the week. This month and next, its "supersaver" package tours are selling "200 percent" ahead of last year. "It's been a banner year for us across the Atlantic."

Business has been so good at Perillo Tours of New York, which specializes in two-week guided tours of Italy, that its five departures a week are booked solid until the last week in October. The firm expects to lead 22,000 Americans to Italy this year, up between 3,000 and 4,000. "It's a very, very good year," says manager Margaret Toupin.

Trusthouse Forte and Thistle, two large British hotel chains, also report significantly better business this year. "We're doing very well," says Michael Gray of Trusthouse Forte's New York sales office.

Despite the demand for rooms, tourist offices and travel agents agree that space is available, although some tourists may not get their first choice.

Generally, "there's no problem getting rooms," says Nora Brossard of the French Government Tourist Office, in part because many French families are staying home this summer. But she does advise making hotel reservations in Paris, even through the fall because of anticipated heavy convention and business travel.

Meanwhile, travel authorities are predicting heavier-than-usual fall traffic across the Atlantic, when airfares are reduced even more, and the alpine ski capitals are hoping for an equally bright winter. The fall attractions include popular wine and harvest festivals and the opening of the new music and theater seasons.

British Airways says its "forward" bookings into September and October are running 75 percent higher than last year. TWA, which normally halts its Boston-to-Rome flights on the first of October, has extended them through the month because of the demand. Air France reports its September advance bookings running 13 percent ahead. At Trusthouse Forte hotels, "our bookings are very, very high for September and October," says Gray. "It looks like an extremely strong fall and winter."

Despite the crowds, the feeling among travel officials is that most of the visitors are enjoying themselves. And one thing never changes: First-time travelers abroad still get their kicks from both the exotic and the mundane. In a fancy Paris restaurant near the Eiffel Tower, a tour group from Michigan could be heard earlier this month hooting at France's waxpaper-like version of toilet paper.

"Wait 'til my friends back home see this," said one of them, stuffing a handful of the slick tissues into her purse.