The 19-year-old midfielder agreed a four-year deal with Celtic after initially rejecting the chance to join them when Motherwell accepted a £3.25 million offer last week.
Turnbull is Celtic’s first new recruit ahead of the 2019-20 campaign, which will begin with the first leg of their opening Champions League qualifier against Sarajevo in Bosnia on 9 July. Lennon anticipates another two arrivals before the 4 July Uefa signing deadline for that tie with further additions targeted thereafter.
He declined to comment on Celtic being linked with 28-year-old Danish international right-back Peter Ankersen of Copenhagen, a potential replacement for Mikael Lustig, who has joined Belgian club Gent, but admitted defensive reinforcements are among his priorities.
“We are looking to bring in maybe half a dozen players in this window,” said Lennon.
“We are certainly looking to strengthen in defensive areas. We are hoping to get two or three in before the first Champions League deadline. Hopefully we can get them out to our pre-season training camp in Austria next week and integrate them into the squad as quickly as possible.
“It’s an important window because we have priorities – the league and hopefully also Champions League qualification, which would set the season up for us. We are working very hard to find the right quality of player within our wage bracket to make us better. We also have to be aware there may be bids for one or two of our players. Some players may leave and we are on top of that as well.”