The objective of this research was to find out how the two search engines Google and Bing perform when users work freely on pre-defined tasks, and judge the relevance of the results immediately after finishing their search session. In a user study, 64 participants conducted two search tasks each, and then judged the results on the following: (1) The quality of the results they selected in their search sessions, (2) The quality of the results they were presented with in their search sessions (but which they did not click on), (3) The quality of the results from the competing search engine for their queries (which they did not see in their search session). We found that users heavily relied on Google, that Google produced more relevant results than Bing, that users were well able to select relevant results from the results lists, and that users judged the relevance of results lower when they regarded a task as difficult and did not find the correct information.