This paper presents an approach to the challenging issue of passive source localization in shallow water using a mobile short horizontal linear array with length less than ten meters. The short array can be conveniently placed on autonomous underwater vehicles and deployed for adaptive spatial sampling. However, the use of such small aperture passive sonar systems makes it difficult to acquire sufficient spatial gain for localizing long-range sources. To meet the requirement, a localization approach that employs matched-field based techniques that enable the short horizontal linear array is used to passively localize acoustic sources in shallow water. Furthermore, the broadband processing and inter-position processing provide robustness against ocean environmental mismatch and enhance the stability of the estimation process. The proposed approach's ability to localize acoustic sources in shallow water at different signal-to-noise ratios is examined through the synthetic test cases where the sources are located at the endfire and some other bearing of the mobile short horizontal linear array. The presented results demonstrate that the positional parameters of the estimated source build up over time as the array moves at a low speed along a straight line at a constant depth.