It may be thousands of miles to travel and the cost of aviation fuel is not exactly cheap, but there are ways to get to European destinations from America without it breaking the bank. These depend on the timing of the flights and your flexibility. Let's start off with your choice of the time to make the trip. If you pick the high season, you will pay a premium. That's not a little amount. There can be more than a thousand dollars difference in the price of a ticket depending on the route and the airline. For these purposes, the price period to avoid is June through August. This reflects the weather pattern usually expected in Europe. After what can be freezing temperatures over winter and spring, we warm up through April and May, and get steady heat June through August. If you go in April and May, or postpone the trip into the fall, the most popular tourist destinations are less full of visitors so the hotel rooms cost less and the weather is still inviting.
Of course, if you grew up in a cold climate or you are going for the skiing, the cheapest periods are either side of Christmas from November through March. Why are you going on this trip? If it is just for rest and relaxation, good food and alcohol when in the mood, it does not matter when you go and you can take advantage of the cheap fights off-season. Then as to flexibility, use the search engine to explore the prices months in advance. Fares are cheaper at some times and on some days rather than others. Only by making multiple searches can you begin to see the pattern and adjust your travel plans to take advantage of the cheap flights. For some European destinations, it can save money by picking routes where your intended destination is only a refueling stop, e.g. some New York flights to India stop in London. Or break your journey, e.g. a flight to Rome can stop in either Madrid or Brussels. The more you are prepared to accept different options, the lower the end-price is likely to be on your cheap flights.