The Scot could not get the better of the Kazakh, whose quirky style, big serve and excellent hand skills were too strong for the 34-year-old, world No 88, prevailing 7-6 (11/9) 6-3 in just under two hours.
Murray will regret not taking a set point on his serve in a very close opening set, while Bublik secured a crucial break in the second set to set up a date with either Tommy Paul or Grigor Dimitrov in the third round.
Emma Raducanu’s campaign in Indian Wells also ended in bitter disappointment after she suffered a painful lost to Petra Martic.
The British No 1, seeded 11 for the event in the Californian desert, served for the match in a topsy-turvy encounter with the Croatian, but could not get over the finishing line against the world No 79.
Raducanu was a warm favourite to reach the last 16. She took the first set on tie-break, but lost the second 6-4 and appeared to struggle physically with the hot conditions.
However, she gave herself an excellent chance to progress past the third round when she broke for a 5-4 lead in the final set, only to be broken right back and then going on to lose the next three games, with Martic claiming a 6-7 (3/7) 6-4 7-5 victory in two hours and 49 minutes.
Martic goes on to face 28th seed Ludmilla Samsonova of Russia in the next round.
There was also a sore defeat for Britain’s Jamie Murray and Bruno Soares in the men’s doubles as they went down 7-6 (7/4) 6-4 to Rohan Bopanna and Denis Shapovalov in the first round.
Murray and Soares have won major doubles titles together before, but the Scot and his Brazilian counterpart were overpowered by the experience and nous of the India-Canada duo.
British hopes continue to be represented in the men’s draw by Cameron Norrie and Dan Evans, who are both in third-round action on Monday in Indian Wells.
Norrie, the defending champion, takes on Georgian Nikoloz Basilashvili, while Evans has an altogether tougher assignment against Rafael Nadal, who has yet to taste defeat this year.
Meanwhile, Naomi Osaka was reduced to tears after being heckled during her second-round defeat to Veronika Kudermetova.
The Japanese player, who missed parts of the 2021 season because of mental health issues, was jeered in the early part of the match and it was undoubtedly a major factor as she lost 6-0 6-4 to the world No 24.
A spectator reportedly shouted "Naomi, you suck", with Osaka complaining to the umpire. As she went to serve in the third game of the match she was visibly crying.
Clearly affected, she lost the first set without winning a game, before putting up a better fight in the second set.
Osaka made the unconventional decision to stay on court and be interviewed after her loss where she said the incident reminded her of a video she watched of Venus and Serena Williams receiving abuse at the same venue in 2001.
The Williams sisters boycotted the event – widely regarded to be the biggest outside of the four grand slams – for 14 years after being racially abused 21 years ago.
Speaking through tears on court, Osaka said: "I just wanted to say thank you, I feel like I cry enough on camera.
"To be honest, I've been heckled before, it doesn't really bother me, but being heckled here, I watched a video of Venus and Serena getting heckled here, if you've never watched it you should watch it and I don't know why, but it got into my head and it got replayed a lot."