Where I am (a Canadian municipality) transit and other municipal-led projects have several features that tend to keep them on budget and schedule (or at least closer to it):

- Unit price contracts (much easier to track growth in cost, helps to identify and mitigate issues earlier)

- Vendor performance management (Contractors and consultants are given a score after project completion, shapes how bids are evaluated on future projects)

- Qualified bidders (contractors and consultants are evaluated beforehand, only allowed to bid on projects if they've done something of a similar scope/content before - ties in with the point above, smaller projects will allow bidding by contractors with less history/lower scores)

The system isn't perfect (since it relies on vendor history, it only works well with projects of scope and type that they do regularly - i.e. roads and sewers mainly), but it's a lot better than a simplistic "lowest bid wins"