I have a quite simple animation, a text in a big font moving continuously (pixel by pixel) to the left. The text is first converted to an image, then a timer task is started
Profiling shows that you are saturating the shared thread used by javax.swing.Timer. One mitigation strategy is to use a longer period and/or a larger increment/decrement, as shown here.
Addendum: In addition, you are laboriously re-rendering the entire image in each call to paintComponent(). Instead, render it once using TextLayout, seen here, and draw() only the newly visible portion each time.

Try calling this method when you are done drawing:
Toolkit.getDefaultToolkit().sync();
This flushs the graphics buffer which some systems like Linux use. See the Javadoc: http://docs.oracle.com/javase/7/docs/api/java/awt/Toolkit.html#sync()
getGraphics and you should NEVER call paintComponent yourself, this not how custom painting is done in Swing. Instead, update the state and call repaint. getWidth and getHeight)super.paintComponent. This is even more important with JComponent, as it is not opaque and failing to do so could result in some nasty paint artefacts. JComponent#getPreferredSize so it can work with layout managers for efficiently.Take a look at Swing animation running extremely slow, which through some object management and optimisation, was able to increase from 500 animated objects up to 4500.
Also take a look at Performing Custom Painting and Painting in AWT and Swing in particular
To answer my own question: After realizing that any continuous input (mouse or keyboard) makes the animation run smoothly, I remembered that inputs can be generated by the program itself, using an object of the class java.awt.Robot. That lead to a simple workaround:
Create a Robot and let it press a key or a mouse move in each animation cycle.
final Robot robot = new Robot();
javax.swing.Timer timer = new javax.swing.Timer(initialDelay, new ActionListener()
{
public void actionPerformed(ActionEvent e)
{
// update your image...
robot.keyPress(62);
}
});
This is a kludge, but works perfectly.