Conversational search provides users with a natural and convenient new search experience. Recently, conversational dense retrieval has shown to be a promising technique for realizing conversational search. However, as conversational search systems have not been widely deployed, it is hard to get large-scale real conversational search sessions and relevance labels to support the training of conversational dense retrieval. To tackle this data scarcity problem, previous methods focus on developing better few-shot learning approaches or generating pseudo relevance labels, but the data they use for training still heavily rely on manual generation.In this paper, we present ConvTrans, a data augmentation method that can automatically transform easily-accessible web search sessions into conversational search sessions to fundamentally alleviate the data scarcity problem for conversational dense retrieval. ConvTrans eliminates the gaps between these two types of sessions in terms of session quality and query form to achieve effective session transformation. Extensive evaluations on two widely used conversational search benchmarks, i.e., CAsT-19 and CAsT-20, demonstrate that the same model trained on the data generated by ConvTrans can achieve comparable retrieval performance as it trained on high-quality but expensive artificial conversational search data.