Slightly old info, but:
SEC filings for end of 2016 had Visa circulation at 335 million, MC at 200 million, Discover at 51.4 and Amex at 47.5.
https://wallethub.com/edu/market-share-by-credit-card-networ...
Nilson report 2016 card usage by volume has Visa at $1.549 trillion, with Amex at $0.695 trillion, Mastercard at $0.693 trillion.
https://www.valuepenguin.com/largest-credit-card-issuers
Forbes has some charts with market share info.
https://www.forbes.com/sites/greatspeculations/2017/09/07/vi...
Moreover, that still wouldn't give you competition, because then what incentive does Visa have to give you a good rate if they're the only network you could plausibly attempt exclusivity with? It's like trying to solve a lack of competition by creating a monopoly.