Participants in matching markets face search and screening costs which prevent the market from clearing efficiently. In many settings, the rise of online matching platforms has dramatically reduced the cost of finding and contacting potential partners. While one might expect both sides of the market to benefit from reduced search costs, this is far from guaranteed. In particular, this change may force participants to screen more potential partners before finding one who is willing to accept their offer. Thus, for a fixed screening cost, reductions in search and application costs may actually decrease aggregate welfare. We illustrate this fact using a model in which agents apply and screen asynchronously. In our model, it is possible that an agent on one side of the market (an "employer") identifies a suitable match on the other side (an "applicant"), only to find that this applicant has already matched. We find that equilibrium is generically inefficient for both employers and applicants. Most notably, when application costs are sufficiently small, uncertainty about applicant availability may drive equilibrium employer welfare to zero. We consider a simple intervention available to the platform: limiting the visibility of applicants. We find that this intervention can significantly improve the welfare of agents on both sides of the market; applicants pay lower application costs, while employers are less likely to find that the applicants they screen have already matched. Somewhat counterintuitively, the benefits of showing fewer applicants to each employer are greatest in markets in which there is a shortage of applicants.