As ever things are quite complicated.
TfL are a local government body responsible for Transport in London - buses, some roads (red routes), DLR, Underground, trams and a limited number of overground services (eg Overground, TfL Rail). The Mayor and GLA have control over those.
However a lot of mainline stations and overland track is owned by another public body - Network Rail which the Mayor and GLA have no control over - which for example is why the Overground 24 service does not come further than New Cross as it runs on London Underground lines until then and Network Rail lines after that.
Finally a lot of London, especially SE London, is serviced by overground trains running in to mainline stations. These are run by private train operating companies (TOCs) whose franchises are awarded and monitored by the Department of Transport. Again the Mayor and GLA have no control over those.
In terms of “bailouts” I understand that yesterdays £1.6 bn was just for TfL.
In terms of the TOCs franchises have already been suspended which essentially means government is covering their operating costs now.