In recent years, customized buses (CBs), a new form of public travel mode between bus and car, has sprung up in China. Its characteristics include flexible routes, each person having a seat and point-to-point travel have attracted travelers who seek high-quality travel, especially car travelers, alleviating traffic congestion at peak periods and leading to a change in urban travel modes. In addition to providing new travel modes, an exclusive bus lane (EBL) is also an effective means to alleviate traffic congestion. Therefore, this paper establishes link impedance functions under mixed travel modes considering the EBL, including customized buses on different kinds of links, and then presents a day-to-day dynamic traffic flow assignment model based on stochastic user equilibrium (SUE). Some conclusions were summarized by numerical case studies. First, the parameter of travelers’ sensitivity to route travel time affects the speed of traffic flow evolution. When it increases to positive infinity, the final state of the traffic network moves from SUE to deterministic user equilibrium (DUE). Second, the parameter on the degree of dependence of travelers on previous experience can not only influence the value size of actual travel time, but also influence the direction of actual travel time evolution. Third, conventional buses and customized buses have higher transportation efficiency than cars, but if the proportion of conventional bus travelers is too large, the total travel time of all travelers in the traffic network may increase. Fourth, when travel demands increase, the proportion of travelers who choose public transit is required to increase to achieve minimum total travel time. Lastly, from the perspective of the whole traffic network in any case, the EBL is not always beneficial. It is recommended to set EBLs when conventional buses and customized bus flows are heavy, which can be judged based on the model established in this paper.