In some ways, one can argue that the New Deal never ended. Many aspects of the New Deal became permanent parts of our society. We still, for example, have Social Security and the FDIC and other programs that either started in the New Deal or were created (like Medicare) to expand on the ideas of the New Deal.
If you want to say that the New Deal ended in the late '30s, though, there are two main reasons why it did.
First, there was the fact that many in Congress and many in conservatives in general were starting to push back against the New Deal. In 1938, for example, Republicans made significant gains in the Congressional elections, showing that there was significant anti-New Deal feeling in the country.
Second, the coming of crises and eventually war in Europe and Asia took the attention of the country and the government away from domestic programs. As the Roosevelt Administration had to concentrate more on foreign affairs, it stopped pushing so hard for domestic reforms.
If you are going to argue that the New Deal ended in the '30s, these are the most important reasons why it did.
Packing” the Supreme Court. The key element in Roosevelt's proposal to reorganize the federal judiciary focused on the Supreme Court. The Court had already invalidated two major pieces of New Deal legislation, the AAA and the NIRA, and other laws were under legal challenge. Roosevelt wanted the authority to increase the size of the Court from 9 to 15 by appointing one new justice for each justice over the age of 70 who did not retire. This power would have allowed him to immediately appoint members who were sympathetic to the New Deal. The ploy fooled no one, and Congress, many Democrats included, rejected the proposal at the cost of considerable political capital for Roosevelt. Ironically, the president got what he wanted anyway. The Court upheld a number of programs of the Second New Deal, including the Social Security and Wagner Acts, and retirements and deaths allowed Roosevelt to appoint his own justices: Hugo Black, Felix Frankfurter, and William O. Douglas.
The recession of 1937. Although unemployment remained high, the economy had steadily improved from 1933 through the first half of 1937. In the late summer of 1937, however, the country entered a deep recession that lasted almost a year. This major slump was caused by the sharp cuts in federal spending that the administration thought were necessary to control the growing deficit and by a reduction in disposable income due to Social Security payroll taxes. Industrial production declined, the number of people out of work grew, and stock prices fell. By the spring, Roosevelt reversed course and called on Congress to pass a massive public works spending program. The Emergency Relief Appropriation Act (June 1938) created more jobs through the WPA and made more money for direct relief and government loans available. At the same time, the Federal Reserve Board adopted an easier credit policy.
Roosevelt's attempt to “pack” the Supreme Court combined with the recession to undercut the New Deal. Politically, Roosevelt faced a coalition of Republicans and conservative Democrats willing to exercise its muscle. When the president called Congress into special session in the fall of 1937 to enact a broad range of legislation, including the reorganization of the executive branch, the coalition prevented the passage of all the bills. The two major accomplishments during this period of the president's second term were the Second Agricultural Adjustment Act (February 1938) and the Fair Labor Standards Act, also known as the Wages and Hours Act (June 1938). The Second AAA provided for the storage of surplus crops in government warehouses and made loans to farmers in years of overproduction to compensate for lower market prices. The Wages and Hours Act phased in minimum wage and maximum hour (40 hours per week) requirements for businesses that engaged in or were affected by interstate commerce, and provided time-and-a-half for overtime work. The statute also prohibited labor by children under the age of 16 and restricted those under 18 to non-hazardous work. Although the Democrats retained their control of Congress in 1938, the Republicans gained seats in the House and the Senate for the first time since 1928. In the wake of the elections, Roosevelt did not offer any new domestic programs in his State of the Union address (January 1939) but focused instead on the threat that aggressor nations posed to international peace.