Sinclair, 20, rejected a new Liverpool deal to join Watford on a five-year contract in June for a compensation fee of about £4m.
He scored his first Hornets goal in the FA Cup win over Burton on 7 January.
Polish centre-back Bielik, 19, joined Arsenal from Legia Warsaw for £2.4m in 2015 but has made just two appearances.
Both games came in EFL Cup ties against Championship opposition, the most recent a 4-0 win at Nottingham Forest in September.
Find all the latest football transfers on our dedicated page or visit our Premier League tracker here.
Birmingham City have signed Watford striker Jerome Sinclair and Arsenal defender Krystian Bielik on loan until the end of the season.